What is the integral of (x^3)/((1+5x^2)^3) ?

The integral of (x^3)/((1+5x^2)^3) is -1/(50(5x^2+1))+1/(100(5x^2+1)^2)+C

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integral of (x^3)/((1+5x^2)^3)

\int \frac{x ^{3}}{( 1 + 5 x ^{2} ) ^{3}} d x

- \frac{1}{50 ( 5 x ^{2} + 1 )} + \frac{1}{100 ( 5 x ^{2} + 1 ) ^{2}} + C

Solution steps

\int \frac{x ^{3}}{( 1 + 5 x ^{2} ) ^{3}} d x

Take the partial fraction of

\frac{x ^{3}}{( 1 + 5 x ^{2} ) ^{3}} : \frac{x}{5 ( 5 x ^{2} + 1 ) ^{2}} - \frac{x}{5 ( 5 x ^{2} + 1 ) ^{3}}

= \int \frac{x}{5 ( 5 x ^{2} + 1 ) ^{2}} - \frac{x}{5 ( 5 x ^{2} + 1 ) ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{5 ( 5 x ^{2} + 1 ) ^{2}} d x - \int \frac{x}{5 ( 5 x ^{2} + 1 ) ^{3}} d x

\int \frac{x}{5 ( 5 x ^{2} + 1 ) ^{2}} d x = - \frac{1}{50 ( 5 x ^{2} + 1 )}

\int \frac{x}{5 ( 5 x ^{2} + 1 ) ^{3}} d x = - \frac{1}{100 ( 5 x ^{2} + 1 ) ^{2}}

= - \frac{1}{50 ( 5 x ^{2} + 1 )} - (- \frac{1}{100 ( 5 x ^{2} + 1 ) ^{2}})

Simplify

= - \frac{1}{50 ( 5 x ^{2} + 1 )} + \frac{1}{100 ( 5 x ^{2} + 1 ) ^{2}}

Add a constant to the solution

= - \frac{1}{50 ( 5 x ^{2} + 1 )} + \frac{1}{100 ( 5 x ^{2} + 1 ) ^{2}} + C

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What is the integral of (x^3)/((1+5x^2)^3) ?
The integral of (x^3)/((1+5x^2)^3) is -1/(50(5x^2+1))+1/(100(5x^2+1)^2)+C