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Resolución de integrales/ x^(2)/ x^(3)/ (x^(2)+1)/(x^(3)+3x+1)^(2)

Integral de (x^(2)+1)/(x^(3)+3x+1)^(2) dx

Solución

Ha introducido [src]

  1                   
  /                   
 |                    
 |        2           
 |       x  + 1       
 |  --------------- dx
 |                2   
 |  / 3          \    
 |  \x  + 3*x + 1/    
 |                    
/                     
0

$$\int\limits_{0}^{1} \frac{x^{2} + 1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}}\, dx$$

Integral((x^2 + 1)/(x^3 + 3*x + 1)^2, (x, 0, 1))

Solución detallada

Hay varias maneras de calcular esta integral.
Método #1
1. Vuelva a escribir el integrando:
  
  $\frac{x^{2} + 1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}} = \frac{x^{2} + 1}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1}$
2. Vuelva a escribir el integrando:
  
  $\frac{x^{2} + 1}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1} = \frac{x^{2}}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1} + \frac{1}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1}$
3. Integramos término a término:
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $\frac{2 x^{2} - x - 1}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t + 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} + \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $- \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
  El resultado es: $\frac{2 x^{2} - x - 1}{15 x^{3} + 45 x + 15} - \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t + 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} + \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
Método #2
1. Vuelva a escribir el integrando:
  
  $\frac{x^{2} + 1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}} = \frac{x^{2}}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1} + \frac{1}{x^{6} + 6 x^{4} + 2 x^{3} + 9 x^{2} + 6 x + 1}$
2. Integramos término a término:
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $\frac{2 x^{2} - x - 1}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t + 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} + \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $- \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
  El resultado es: $\frac{2 x^{2} - x - 1}{15 x^{3} + 45 x + 15} - \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t + 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} + \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$
Ahora simplificar:

$\frac{\left(x^{3} + 3 x + 1\right) \left(\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right) \log{\left(x - \frac{6}{11} - \frac{4}{165 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{75 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} + \frac{30375 \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right)^{2}}{44} \right)} + \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} - \frac{75 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{4}{165 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} \right)} + \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)} + \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{75 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{6}{11} - \frac{4}{165 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} + \frac{30375 \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} \right)}\right) - \frac{1}{3}}{x^{3} + 3 x + 1}$
Añadimos la constante de integración:

$\frac{\left(x^{3} + 3 x + 1\right) \left(\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right) \log{\left(x - \frac{6}{11} - \frac{4}{165 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{75 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} + \frac{30375 \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right)^{2}}{44} \right)} + \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} - \frac{75 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{4}{165 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} \right)} + \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)} + \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{75 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{6}{11} - \frac{4}{165 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} + \frac{30375 \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} \right)}\right) - \frac{1}{3}}{x^{3} + 3 x + 1}+ \mathrm{constant}$

Respuesta:

$\frac{\left(x^{3} + 3 x + 1\right) \left(\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right) \log{\left(x - \frac{6}{11} - \frac{4}{495 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} - \frac{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} + \frac{30375 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}} + \frac{4}{10125 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}\right)^{2}}{44} \right)} + \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right) \log{\left(x - \frac{6}{11} - \frac{4}{165 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{75 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} + \frac{30375 \left(- \frac{4}{3375 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3}\right)^{2}}{44} \right)} + \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(\frac{4}{10125 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} - \frac{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{4}{495 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} \right)} + \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{6}{11} + \frac{30375 \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} - \frac{75 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{4}{165 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} \right)} + \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right) \log{\left(x - \frac{225 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}}{11} - \frac{6}{11} - \frac{4}{495 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \frac{30375 \left(\frac{4}{10125 \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}} + \sqrt[3]{\frac{44 \sqrt{5}}{2278125} + \frac{4}{91125}}\right)^{2}}{44} \right)} + \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right) \log{\left(x - \frac{75 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{11} - \frac{6}{11} - \frac{4}{165 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}} + \frac{30375 \left(- \frac{\sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}{3} - \frac{4}{3375 \sqrt[3]{\frac{44 \sqrt{5}}{84375} + \frac{4}{3375}}}\right)^{2}}{44} \right)}\right) - \frac{1}{3}}{x^{3} + 3 x + 1}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

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 |       2                                2                   2            /                                /                          2\\          /                                /                          2\\
 |      x  + 1                -1 - x + 2*x         4 - x + 2*x             |       3                        |  6        225*t   30375*t ||          |       3                        |  6        225*t   30375*t ||
 | --------------- dx = C + ----------------- - ----------------- + RootSum|91125*t  - 108*t - 8, t -> t*log|- -- + x - ----- + --------|| + RootSum|91125*t  - 108*t + 8, t -> t*log|- -- + x + ----- + --------||
 |               2                   3                   3                 \                                \  11         11       44   //          \                                \  11         11       44   //
 | / 3          \           15 + 15*x  + 45*x   15 + 15*x  + 45*x                                                                                                                                                  
 | \x  + 3*x + 1/                                                                                                                                                                                                  
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$$\int \frac{x^{2} + 1}{\left(\left(x^{3} + 3 x\right) + 1\right)^{2}}\, dx = C + \frac{2 x^{2} - x - 1}{15 x^{3} + 45 x + 15} - \frac{2 x^{2} - x + 4}{15 x^{3} + 45 x + 15} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t + 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} + \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)} + \operatorname{RootSum} {\left(91125 t^{3} - 108 t - 8, \left( t \mapsto t \log{\left(\frac{30375 t^{2}}{44} - \frac{225 t}{11} + x - \frac{6}{11} \right)} \right)\right)}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

4/15

$$\frac{4}{15}$$

4/15

$$\frac{4}{15}$$

4/15

Respuesta numérica [src]

0.266666666666667

0.266666666666667

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

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Expresiones idénticas

Expresiones semejantes

Integral de (x^(2)+1)/(x^(3)+3x+1)^(2) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución

Método #1

Método #2