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

The integral of (x+5)/((x^2+5)^3) is

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

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

- \frac{1}{4 ( x ^{2} + 5 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 3 5 x ^{3} + 30 x ^{2} arctan ( \frac{x}{5} ) + 25 5 x + 75 arctan ( \frac{x}{5} )}{40 5 ( x ^{2} + 5 ) ^{2}} + C

Solution steps

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

Expand

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

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

\int \frac{x}{( x ^{2} + 5 ) ^{3}} d x = - \frac{1}{4 ( x ^{2} + 5 ) ^{2}}

\int \frac{5}{( x ^{2} + 5 ) ^{3}} d x = \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 3 5 x ^{3} + 30 x ^{2} arctan ( \frac{x}{5} ) + 25 5 x + 75 arctan ( \frac{x}{5} )}{40 5 ( x ^{2} + 5 ) ^{2}}

= - \frac{1}{4 ( x ^{2} + 5 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 3 5 x ^{3} + 30 x ^{2} arctan ( \frac{x}{5} ) + 25 5 x + 75 arctan ( \frac{x}{5} )}{40 5 ( x ^{2} + 5 ) ^{2}}

Add a constant to the solution

= - \frac{1}{4 ( x ^{2} + 5 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 3 5 x ^{3} + 30 x ^{2} arctan ( \frac{x}{5} ) + 25 5 x + 75 arctan ( \frac{x}{5} )}{40 5 ( x ^{2} + 5 ) ^{2}} + C

\frac{\partial}{\partial x} (x^{6} y - 3 x^{5} y^{2})

\frac{\partial}{\partial x} (x (3 - 2 x - y))

\int e^{x} + 5 d x

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