What is the integral of (x^2)/((x^2+9)^3) ?

The integral of (x^2)/((x^2+9)^3) is

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integral of (x^2)/((x^2+9)^3)

\int \frac{x ^{2}}{( x ^{2} + 9 ) ^{3}} d x

\frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))) - \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{72 ( x ^{2} + 9 ) ^{2}} + C

Solution steps

\int \frac{x ^{2}}{( x ^{2} + 9 ) ^{3}} d x

Take the partial fraction of

\frac{x ^{2}}{( x ^{2} + 9 ) ^{3}} : \frac{1}{( x ^{2} + 9 ) ^{2}} - \frac{9}{( x ^{2} + 9 ) ^{3}}

= \int \frac{1}{( x ^{2} + 9 ) ^{2}} - \frac{9}{( x ^{2} + 9 ) ^{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{1}{( x ^{2} + 9 ) ^{2}} d x - \int \frac{9}{( x ^{2} + 9 ) ^{3}} d x

\int \frac{1}{( x ^{2} + 9 ) ^{2}} d x = \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3})))

\int \frac{9}{( x ^{2} + 9 ) ^{3}} d x = \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{72 ( x ^{2} + 9 ) ^{2}}

= \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))) - \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{72 ( x ^{2} + 9 ) ^{2}}

Add a constant to the solution

= \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))) - \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{72 ( x ^{2} + 9 ) ^{2}} + C

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What is the integral of (x^2)/((x^2+9)^3) ?
The integral of (x^2)/((x^2+9)^3) is