What is the integral of (9x^2-26x-5)/(3x^2-5x-2) ?

The integral of (9x^2-26x-5)/(3x^2-5x-2) is

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integral of (9x^2-26x-5)/(3x^2-5x-2)

\int \frac{9 x ^{2} - 26 x - 5}{3 x ^{2} - 5 x - 2} d x

3 x - 22 (\frac{1}{12} ln 36 x^{2} - 60 x - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x) + C

Solution steps

\int \frac{9 x ^{2} - 26 x - 5}{3 x ^{2} - 5 x - 2} d x

Long division

\frac{9 x ^{2} - 26 x - 5}{3 x ^{2} - 5 x - 2} : 3 + \frac{- 11 x + 1}{3 x ^{2} - 5 x - 2}

= \int 3 + \frac{- 11 x + 1}{3 x ^{2} - 5 x - 2} 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 3 d x + \int \frac{- 11 x + 1}{3 x ^{2} - 5 x - 2} d x

\int 3 d x = 3 x

\int \frac{- 11 x + 1}{3 x ^{2} - 5 x - 2} d x = - 22 (\frac{1}{12} ln 36 x^{2} - 60 x - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)

= 3 x - 22 (\frac{1}{12} ln 36 x^{2} - 60 x - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)

Add a constant to the solution

= 3 x - 22 (\frac{1}{12} ln 36 x^{2} - 60 x - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} x - ln - \frac{12}{7} + \frac{6}{7} x) + C

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What is the integral of (9x^2-26x-5)/(3x^2-5x-2) ?
The integral of (9x^2-26x-5)/(3x^2-5x-2) is