What is the integral of (x^3+2x)/(x^2+x-6) ?

The integral of (x^3+2x)/(x^2+x-6) is

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integral of (x^3+2x)/(x^2+x-6)

\int \frac{x ^{3} + 2 x}{x ^{2} + x - 6} d x

\frac{x ^{2}}{2} - x + \frac{9}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{5} ln \frac{2}{5} x + \frac{6}{5} - \frac{7}{5} ln \frac{2}{5} x - \frac{4}{5} + \frac{7}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

Solution steps

\int \frac{x ^{3} + 2 x}{x ^{2} + x - 6} d x

Expand

\frac{x ^{3} + 2 x}{x ^{2} + x - 6} : \frac{x ^{3}}{x ^{2} + x - 6} + \frac{2 x}{x ^{2} + x - 6}

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 ^{3}}{x ^{2} + x - 6} d x + \int \frac{2 x}{x ^{2} + x - 6} d x

\int \frac{x ^{3}}{x ^{2} + x - 6} d x = \frac{x ^{2}}{2} - x + 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

\int \frac{2 x}{x ^{2} + x - 6} d x = ln 4 x^{2} + 4 x - 24 + \frac{1}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= \frac{x ^{2}}{2} - x + 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln 4 x^{2} + 4 x - 24 + \frac{1}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

Simplify

\frac{x ^{2}}{2} - x + 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln 4 x^{2} + 4 x - 24 + \frac{1}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) : \frac{x ^{2}}{2} - x + \frac{9}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{5} ln \frac{2}{5} x + \frac{6}{5} - \frac{7}{5} ln \frac{2}{5} x - \frac{4}{5} + \frac{7}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= \frac{x ^{2}}{2} - x + \frac{9}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{5} ln \frac{2}{5} x + \frac{6}{5} - \frac{7}{5} ln \frac{2}{5} x - \frac{4}{5} + \frac{7}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

Add a constant to the solution

= \frac{x ^{2}}{2} - x + \frac{9}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{5} ln \frac{2}{5} x + \frac{6}{5} - \frac{7}{5} ln \frac{2}{5} x - \frac{4}{5} + \frac{7}{10} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

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What is the integral of (x^3+2x)/(x^2+x-6) ?
The integral of (x^3+2x)/(x^2+x-6) is