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3x^2+2x/5x^5+8
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Expresiones semejantes
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Resolución de integrales/ x^(2)/ (5*x^(5)+8)/ (3*x^(2)+2*x)/(5*x^(5)+8)

Integral de (3x^(2)+2x)/(5*x^(5)+8) dx

Solución

Ha introducido [src]

 oo              
  /              
 |               
 |     2         
 |  3*x  + 2*x   
 |  ---------- dx
 |      5        
 |   5*x  + 8    
 |               
/                
2

$$\int\limits_{2}^{\infty} \frac{3 x^{2} + 2 x}{5 x^{5} + 8}\, dx$$

Integral((3*x^2 + 2*x)/(5*x^5 + 8), (x, 2, oo))

Solución detallada

Hay varias maneras de calcular esta integral.
Método #1
1. Vuelva a escribir el integrando:
  
  $\frac{3 x^{2} + 2 x}{5 x^{5} + 8} = \frac{x \left(3 x + 2\right)}{5 x^{5} + 8}$
2. Vuelva a escribir el integrando:
  
  $\frac{x \left(3 x + 2\right)}{5 x^{5} + 8} = \frac{3 x^{2}}{5 x^{5} + 8} + \frac{2 x}{5 x^{5} + 8}$
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:
    
    $\int \frac{3 x^{2}}{5 x^{5} + 8}\, dx = 3 \int \frac{x^{2}}{5 x^{5} + 8}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)}$
    Por lo tanto, el resultado es: $3 \operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \frac{2 x}{5 x^{5} + 8}\, dx = 2 \int \frac{x}{5 x^{5} + 8}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
    Por lo tanto, el resultado es: $2 \operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
  El resultado es: $3 \operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)} + 2 \operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
Método #2
1. Vuelva a escribir el integrando:
  
  $\frac{3 x^{2} + 2 x}{5 x^{5} + 8} = \frac{3 x^{2}}{5 x^{5} + 8} + \frac{2 x}{5 x^{5} + 8}$
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:
    
    $\int \frac{3 x^{2}}{5 x^{5} + 8}\, dx = 3 \int \frac{x^{2}}{5 x^{5} + 8}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)}$
    Por lo tanto, el resultado es: $3 \operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \frac{2 x}{5 x^{5} + 8}\, dx = 2 \int \frac{x}{5 x^{5} + 8}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
    Por lo tanto, el resultado es: $2 \operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
  El resultado es: $3 \operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)} + 2 \operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$
Ahora simplificar:

$\frac{3 \cdot 20^{\frac{2}{5}} \log{\left(x + \frac{\sqrt[5]{5000}}{5} \right)}}{100} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 2 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 40000 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 2 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)}$
Añadimos la constante de integración:

$\frac{3 \cdot 20^{\frac{2}{5}} \log{\left(x + \frac{\sqrt[5]{5000}}{5} \right)}}{100} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 2 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 40000 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 2 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)}+ \mathrm{constant}$

Respuesta:

$\frac{3 \cdot 20^{\frac{2}{5}} \log{\left(x + \frac{\sqrt[5]{5000}}{5} \right)}}{100} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{20^{\frac{2}{5}}}{400} + \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100}\right)^{2} \right)} + 2 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x - 40000 \left(\frac{\sqrt[5]{250}}{400} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{3} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} - \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 3 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right) \log{\left(x + 1000 \left(- \frac{2^{\frac{4}{5}} \cdot 5^{\frac{9}{10}}}{400} - \frac{20^{\frac{2}{5}}}{400} + \frac{20^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{100}\right)^{2} \right)} + 2 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{11 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{5^{\frac{7}{10}} \sqrt[5]{50}}{800} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{9 \sqrt[5]{250}}{1600} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{80} - \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{200}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} + \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80} - \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400}\right)^{3} \right)} + 2 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right) \log{\left(x - 40000 \left(- \frac{\sqrt[5]{250} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{100} - \frac{\sqrt[5]{2} \sqrt[10]{5}}{320} - \frac{\sqrt[5]{250}}{1600} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}}}{1600} + \frac{\sqrt[5]{2} \cdot 5^{\frac{3}{5}}}{320} - \frac{\sqrt[5]{250} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{250} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{400} + \frac{\sqrt[5]{10} \cdot 5^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{400} + \frac{\sqrt[5]{2} \sqrt[10]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{80}\right)^{3} \right)}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

  /                                                                                                                              
 |                                                                                                                               
 |    2                                                                                                                          
 | 3*x  + 2*x                   /          5                /           3\\            /          5                /          2\\
 | ---------- dx = C + 2*RootSum\40000000*t  + 1, t -> t*log\x - 40000*t // + 3*RootSum\25000000*t  - 1, t -> t*log\x + 1000*t //
 |     5                                                                                                                         
 |  5*x  + 8                                                                                                                     
 |                                                                                                                               
/

$$\int \frac{3 x^{2} + 2 x}{5 x^{5} + 8}\, dx = C + 3 \operatorname{RootSum} {\left(25000000 t^{5} - 1, \left( t \mapsto t \log{\left(1000 t^{2} + x \right)} \right)\right)} + 2 \operatorname{RootSum} {\left(40000000 t^{5} + 1, \left( t \mapsto t \log{\left(- 40000 t^{3} + x \right)} \right)\right)}$$

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Integral de (3x^(2)+2x)/(5*x^(5)+8) dx

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Integral de (3*x^(2)+2*x)/(5*x^(5)+8) dx

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Solución

Método #1

Método #2

Integral de (3x^(2)+2x)/(5*x^(5)+8) dx