What is the integral of (x^4)/((x^3+x)(x^2-x+3)) ?

The integral of (x^4)/((x^3+x)(x^2-x+3)) is

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integral of (x^4)/((x^3+x)(x^2-x+3))

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

- \frac{1}{5} ln x^{2} + 1 + \frac{1}{5} arctan (x) + \frac{7}{10} ln 4 x^{2} - 4 x + 12 + \frac{1}{5 11} arctan (\frac{1}{11} (2 x - 1)) + C

Solution steps

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

Take the partial fraction of

\frac{x ^{4}}{( x ^{3} + x ) ( x ^{2} - x + 3 )} : \frac{- 2 x + 1}{5 ( x ^{2} + 1 )} + \frac{7 x - 3}{5 ( x ^{2} - x + 3 )}

= \int \frac{- 2 x + 1}{5 ( x ^{2} + 1 )} + \frac{7 x - 3}{5 ( x ^{2} - x + 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{- 2 x + 1}{5 ( x ^{2} + 1 )} d x + \int \frac{7 x - 3}{5 ( x ^{2} - x + 3 )} d x

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

\int \frac{7 x - 3}{5 ( x ^{2} - x + 3 )} d x = \frac{1}{10 11} (711 ln 4 x^{2} - 4 x + 12 + 2 arctan (\frac{2 x - 1}{11}))

= \frac{1}{5} (- ln x^{2} + 1 + arctan (x)) + \frac{1}{10 11} (711 ln 4 x^{2} - 4 x + 12 + 2 arctan (\frac{2 x - 1}{11}))

Simplify

\frac{1}{5} (- ln x^{2} + 1 + arctan (x)) + \frac{1}{10 11} (711 ln 4 x^{2} - 4 x + 12 + 2 arctan (\frac{2 x - 1}{11})) : - \frac{1}{5} ln x^{2} + 1 + \frac{1}{5} arctan (x) + \frac{7}{10} ln 4 x^{2} - 4 x + 12 + \frac{1}{5 11} arctan (\frac{1}{11} (2 x - 1))

= - \frac{1}{5} ln x^{2} + 1 + \frac{1}{5} arctan (x) + \frac{7}{10} ln 4 x^{2} - 4 x + 12 + \frac{1}{5 11} arctan (\frac{1}{11} (2 x - 1))

Add a constant to the solution

= - \frac{1}{5} ln x^{2} + 1 + \frac{1}{5} arctan (x) + \frac{7}{10} ln 4 x^{2} - 4 x + 12 + \frac{1}{5 11} arctan (\frac{1}{11} (2 x - 1)) + C

\int \frac{2 x + 1}{x ^{2} + x - 4} d x

\int \frac{x ^{3} - 2}{3 x - 3} d x

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

\int \frac{1}{x + 3 + x - 1} d x

\int (1 - 3 x)^{4} d x

What is the integral of (x^4)/((x^3+x)(x^2-x+3)) ?
The integral of (x^4)/((x^3+x)(x^2-x+3)) is