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Resolución de integrales/ x^2+9/ (x^3+6x^2+9)/(5x^6+4x)

Integral de (x^3+6x^2+9)/(5x^6+4x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                 
  /                 
 |                  
 |   3      2       
 |  x  + 6*x  + 9   
 |  ------------- dx
 |       6          
 |    5*x  + 4*x    
 |                  
/                   
1
∫1∞(x3+6x2)+95x6+4x dx\int\limits_{1}^{\infty} \frac{\left(x^{3} + 6 x^{2}\right) + 9}{5 x^{6} + 4 x}\, dx 
Integral((x^3 + 6*x^2 + 9)/(5*x^6 + 4*x), (x, 1, oo))
Solución detallada

  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      (x3+6x2)+95x6+4x=−x(45x3−4x−24)4(5x5+4)+94x\frac{\left(x^{3} + 6 x^{2}\right) + 9}{5 x^{6} + 4 x} = - \frac{x \left(45 x^{3} - 4 x - 24\right)}{4 \left(5 x^{5} + 4\right)} + \frac{9}{4 x}

    2. Integramos término a término:

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫(−x(45x3−4x−24)4(5x5+4)) dx=−∫x(45x3−4x−24)5x5+4 dx4\int \left(- \frac{x \left(45 x^{3} - 4 x - 24\right)}{4 \left(5 x^{5} + 4\right)}\right)\, dx = - \frac{\int \frac{x \left(45 x^{3} - 4 x - 24\right)}{5 x^{5} + 4}\, dx}{4}

        1. Vuelva a escribir el integrando:

          x(45x3−4x−24)5x5+4=45x4−4x2−24x5x5+4\frac{x \left(45 x^{3} - 4 x - 24\right)}{5 x^{5} + 4} = \frac{45 x^{4} - 4 x^{2} - 24 x}{5 x^{5} + 4}

        2. Vuelva a escribir el integrando:

          45x4−4x2−24x5x5+4=45x45x5+4−4x25x5+4−24x5x5+4\frac{45 x^{4} - 4 x^{2} - 24 x}{5 x^{5} + 4} = \frac{45 x^{4}}{5 x^{5} + 4} - \frac{4 x^{2}}{5 x^{5} + 4} - \frac{24 x}{5 x^{5} + 4}

        3. Integramos término a término:

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫45x45x5+4 dx=45∫x45x5+4 dx\int \frac{45 x^{4}}{5 x^{5} + 4}\, dx = 45 \int \frac{x^{4}}{5 x^{5} + 4}\, dx

            1. que u=5x5+4u = 5 x^{5} + 4.

              Luego que du=25x4dxdu = 25 x^{4} dx y ponemos du25\frac{du}{25}:

              ∫125u du\int \frac{1}{25 u}\, du

              1. La integral del producto de una función por una constante es la constante por la integral de esta función:

                ∫1u du=∫1u du25\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{25}

                1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

                Por lo tanto, el resultado es: log⁡(u)25\frac{\log{\left(u \right)}}{25}

              Si ahora sustituir uu más en:

              log⁡(5x5+4)25\frac{\log{\left(5 x^{5} + 4 \right)}}{25}

            Por lo tanto, el resultado es: 9log⁡(5x5+4)5\frac{9 \log{\left(5 x^{5} + 4 \right)}}{5}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫(−4x25x5+4) dx=−4∫x25x5+4 dx\int \left(- \frac{4 x^{2}}{5 x^{5} + 4}\right)\, dx = - 4 \int \frac{x^{2}}{5 x^{5} + 4}\, dx

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))\operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)}

            Por lo tanto, el resultado es: −4RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))- 4 \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫(−24x5x5+4) dx=−24∫x5x5+4 dx\int \left(- \frac{24 x}{5 x^{5} + 4}\right)\, dx = - 24 \int \frac{x}{5 x^{5} + 4}\, dx

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

            Por lo tanto, el resultado es: −24RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))- 24 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

          El resultado es: 9log⁡(5x5+4)5−4RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))−24RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\frac{9 \log{\left(5 x^{5} + 4 \right)}}{5} - 4 \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)} - 24 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

        Por lo tanto, el resultado es: −9log⁡(5x5+4)20+RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))+6RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))- \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20} + \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)} + 6 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫94x dx=9∫1x dx4\int \frac{9}{4 x}\, dx = \frac{9 \int \frac{1}{x}\, dx}{4}

        1. Integral 1x\frac{1}{x} es log⁡(x)\log{\left(x \right)}.

        Por lo tanto, el resultado es: 9log⁡(x)4\frac{9 \log{\left(x \right)}}{4}

      El resultado es: 9log⁡(x)4−9log⁡(5x5+4)20+RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))+6RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\frac{9 \log{\left(x \right)}}{4} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20} + \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)} + 6 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

    Método #2

    1. Vuelva a escribir el integrando:

      (x3+6x2)+95x6+4x=x35x6+4x+6x25x6+4x+95x6+4x\frac{\left(x^{3} + 6 x^{2}\right) + 9}{5 x^{6} + 4 x} = \frac{x^{3}}{5 x^{6} + 4 x} + \frac{6 x^{2}}{5 x^{6} + 4 x} + \frac{9}{5 x^{6} + 4 x}

    2. Integramos término a término:

      1. No puedo encontrar los pasos en la búsqueda de esta integral.

        Pero la integral

        RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))\operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫6x25x6+4x dx=6∫x25x6+4x dx\int \frac{6 x^{2}}{5 x^{6} + 4 x}\, dx = 6 \int \frac{x^{2}}{5 x^{6} + 4 x}\, dx

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

        Por lo tanto, el resultado es: 6RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))6 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫95x6+4x dx=9∫15x6+4x dx\int \frac{9}{5 x^{6} + 4 x}\, dx = 9 \int \frac{1}{5 x^{6} + 4 x}\, dx

        1. Vuelva a escribir el integrando:

          15x6+4x=−5x44(5x5+4)+14x\frac{1}{5 x^{6} + 4 x} = - \frac{5 x^{4}}{4 \left(5 x^{5} + 4\right)} + \frac{1}{4 x}

        2. Integramos término a término:

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫(−5x44(5x5+4)) dx=−5∫x45x5+4 dx4\int \left(- \frac{5 x^{4}}{4 \left(5 x^{5} + 4\right)}\right)\, dx = - \frac{5 \int \frac{x^{4}}{5 x^{5} + 4}\, dx}{4}

            1. que u=5x5+4u = 5 x^{5} + 4.

              Luego que du=25x4dxdu = 25 x^{4} dx y ponemos du25\frac{du}{25}:

              ∫125u du\int \frac{1}{25 u}\, du

              1. La integral del producto de una función por una constante es la constante por la integral de esta función:

                ∫1u du=∫1u du25\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{25}

                1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

                Por lo tanto, el resultado es: log⁡(u)25\frac{\log{\left(u \right)}}{25}

              Si ahora sustituir uu más en:

              log⁡(5x5+4)25\frac{\log{\left(5 x^{5} + 4 \right)}}{25}

            Por lo tanto, el resultado es: −log⁡(5x5+4)20- \frac{\log{\left(5 x^{5} + 4 \right)}}{20}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫14x dx=∫1x dx4\int \frac{1}{4 x}\, dx = \frac{\int \frac{1}{x}\, dx}{4}

            1. Integral 1x\frac{1}{x} es log⁡(x)\log{\left(x \right)}.

            Por lo tanto, el resultado es: log⁡(x)4\frac{\log{\left(x \right)}}{4}

          El resultado es: log⁡(x)4−log⁡(5x5+4)20\frac{\log{\left(x \right)}}{4} - \frac{\log{\left(5 x^{5} + 4 \right)}}{20}

        Por lo tanto, el resultado es: 9log⁡(x)4−9log⁡(5x5+4)20\frac{9 \log{\left(x \right)}}{4} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20}

      El resultado es: 9log⁡(x)4−9log⁡(5x5+4)20+RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))+6RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\frac{9 \log{\left(x \right)}}{4} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20} + \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)} + 6 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}

  2. Ahora simplificar:

    9log⁡(x)4+505log⁡(x+50255)50+6(20005400+2025⋅5710400+20005i58−58100)log⁡(x−10000(20005400+2025⋅5710400+20005i58−58100)3)+(−505200+105⋅5710200−505i58+5850)log⁡(x+500(−505200+105⋅5710200−505i58+5850)2)+(−505200+105⋅5710200+505i58+5850)log⁡(x+500(−505200+105⋅5710200+505i58+5850)2)+(−105⋅5710200−505200−505i58−5850)log⁡(x+500(−105⋅5710200−505200−505i58−5850)2)+(−105⋅5710200−505200+505i58−5850)log⁡(x+500(−105⋅5710200−505200+505i58−5850)2)+6(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)log⁡(x−10000(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)3)+6(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)log⁡(x−10000(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)3)+6(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)log⁡(x−10000(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)3)+6(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)log⁡(x−10000(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)3)−9log⁡(5x5+4)20\frac{9 \log{\left(x \right)}}{4} + \frac{\sqrt[5]{50} \log{\left(x + \frac{50^{\frac{2}{5}}}{5} \right)}}{50} + 6 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 10000 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + 6 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 6 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20}

  3. Añadimos la constante de integración:

    9log⁡(x)4+505log⁡(x+50255)50+6(20005400+2025⋅5710400+20005i58−58100)log⁡(x−10000(20005400+2025⋅5710400+20005i58−58100)3)+(−505200+105⋅5710200−505i58+5850)log⁡(x+500(−505200+105⋅5710200−505i58+5850)2)+(−505200+105⋅5710200+505i58+5850)log⁡(x+500(−505200+105⋅5710200+505i58+5850)2)+(−105⋅5710200−505200−505i58−5850)log⁡(x+500(−105⋅5710200−505200−505i58−5850)2)+(−105⋅5710200−505200+505i58−5850)log⁡(x+500(−105⋅5710200−505200+505i58−5850)2)+6(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)log⁡(x−10000(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)3)+6(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)log⁡(x−10000(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)3)+6(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)log⁡(x−10000(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)3)+6(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)log⁡(x−10000(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)3)−9log⁡(5x5+4)20+constant\frac{9 \log{\left(x \right)}}{4} + \frac{\sqrt[5]{50} \log{\left(x + \frac{50^{\frac{2}{5}}}{5} \right)}}{50} + 6 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 10000 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + 6 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 6 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20}+ \mathrm{constant}

Respuesta:

9log⁡(x)4+505log⁡(x+50255)50+6(20005400+2025⋅5710400+20005i58−58100)log⁡(x−10000(20005400+2025⋅5710400+20005i58−58100)3)+(−505200+105⋅5710200−505i58+5850)log⁡(x+500(−505200+105⋅5710200−505i58+5850)2)+(−505200+105⋅5710200+505i58+5850)log⁡(x+500(−505200+105⋅5710200+505i58+5850)2)+(−105⋅5710200−505200−505i58−5850)log⁡(x+500(−105⋅5710200−505200−505i58−5850)2)+(−105⋅5710200−505200+505i58−5850)log⁡(x+500(−105⋅5710200−505200+505i58−5850)2)+6(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)log⁡(x−10000(−11200051600−245⋅535320−59108051600−245510320+2025⋅5710800−5910805i58−58400+245510i58−5880)3)+6(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)log⁡(x−10000(−245⋅535320−2025⋅5710800−59108051600−245510320+9200051600−5910805i58−58400−245510i58−5880−20005i58−58200)3)+6(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)log⁡(x−10000(−245510320−200051600+59108051600+245⋅535320+2000558−5858+58100−245510i58+5880−20005i58+58400−20005i58−58400+5910805i58−58400)3)+6(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)log⁡(x−10000(−2000558−5858+58100−245510320−200051600+59108051600+245⋅535320−20005i58−58400+20005i58+58400+5910805i58−58400+245510i58+5880)3)−9log⁡(5x5+4)20+constant\frac{9 \log{\left(x \right)}}{4} + \frac{\sqrt[5]{50} \log{\left(x + \frac{50^{\frac{2}{5}}}{5} \right)}}{50} + 6 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 10000 \left(\frac{\sqrt[5]{2000}}{400} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} - \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right) \log{\left(x + 500 \left(- \frac{\sqrt[5]{10} \cdot 5^{\frac{7}{10}}}{200} - \frac{\sqrt[5]{50}}{200} + \frac{\sqrt[5]{50} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{50}\right)^{2} \right)} + 6 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{11 \sqrt[5]{2000}}{1600} - \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{20^{\frac{2}{5}} \cdot 5^{\frac{7}{10}}}{800} - \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} + \frac{9 \sqrt[5]{2000}}{1600} - \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 6 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 10000 \left(- \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 6 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 10000 \left(- \frac{\sqrt[5]{2000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{2^{\frac{4}{5}} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{2000}}{1600} + \frac{5^{\frac{9}{10}} \sqrt[5]{80}}{1600} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{5^{\frac{9}{10}} \sqrt[5]{80} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{2^{\frac{4}{5}} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
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 |  3      2                                                                         /       5\                                                             
 | x  + 6*x  + 9                   /         5                /           3\\   9*log\4 + 5*x /   9*log(x)          /         5                /         2\\
 | ------------- dx = C + 6*RootSum\5000000*t  + 1, t -> t*log\x - 10000*t // - --------------- + -------- + RootSum\6250000*t  - 1, t -> t*log\x + 500*t //
 |      6                                                                              20            4                                                      
 |   5*x  + 4*x                                                                                                                                             
 |                                                                                                                                                          
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∫(x3+6x2)+95x6+4x dx=C+9log⁡(x)4−9log⁡(5x5+4)20+RootSum⁡(6250000t5−1,(t↦tlog⁡(500t2+x)))+6RootSum⁡(5000000t5+1,(t↦tlog⁡(−10000t3+x)))\int \frac{\left(x^{3} + 6 x^{2}\right) + 9}{5 x^{6} + 4 x}\, dx = C + \frac{9 \log{\left(x \right)}}{4} - \frac{9 \log{\left(5 x^{5} + 4 \right)}}{20} + \operatorname{RootSum} {\left(6250000 t^{5} - 1, \left( t \mapsto t \log{\left(500 t^{2} + x \right)} \right)\right)} + 6 \operatorname{RootSum} {\left(5000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 10000 t^{3} + x \right)} \right)\right)}
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Gráfica
1.00001.01001.00101.00201.00301.00401.00501.00601.00701.00801.00902.5-2.5
Trazar el gráfico f(x)
Respuesta [src] 
 oo                 
  /                 
 |                  
 |       3      2   
 |  9 + x  + 6*x    
 |  ------------- dx
 |             6    
 |    4*x + 5*x     
 |                  
/                   
1
∫1∞x3+6x2+95x6+4x dx\int\limits_{1}^{\infty} \frac{x^{3} + 6 x^{2} + 9}{5 x^{6} + 4 x}\, dx 
=
= 
 oo                 
  /                 
 |                  
 |       3      2   
 |  9 + x  + 6*x    
 |  ------------- dx
 |             6    
 |    4*x + 5*x     
 |                  
/                   
1
∫1∞x3+6x2+95x6+4x dx\int\limits_{1}^{\infty} \frac{x^{3} + 6 x^{2} + 9}{5 x^{6} + 4 x}\, dx 
Integral((9 + x^3 + 6*x^2)/(4*x + 5*x^6), (x, 1, oo))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

    © Sr Examen – Калькулятор