What is the integral of (45x^2)/(3x^2+x-2) ?

The integral of (45x^2)/(3x^2+x-2) is

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integral of (45x^2)/(3x^2+x-2)

\int \frac{45 x ^{2}}{3 x ^{2} + x - 2} d x

45 (\frac{1}{3} x + \frac{1}{3} (- 2 (\frac{1}{12} ln 36 x^{2} + 12 x - 24 + \frac{ln \frac{6}{5} + \frac{6}{5} x - ln - \frac{4}{5} + \frac{6}{5} x}{60}) + 2 (\frac{1}{5} ln ∣ 3 x - 2 ∣ - \frac{1}{5} ln ∣ x + 1 ∣))) + C

Solution steps

\int \frac{45 x ^{2}}{3 x ^{2} + x - 2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 45 \cdot \int \frac{x ^{2}}{3 x ^{2} + x - 2} d x

Long division

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

= 45 \cdot \int \frac{1}{3} + \frac{- x + 2}{3 ( 3 x ^{2} + 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

= 45 (\int \frac{1}{3} d x + \int \frac{- x + 2}{3 ( 3 x ^{2} + x - 2 )} d x)

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{- x + 2}{3 ( 3 x ^{2} + x - 2 )} d x = \frac{1}{3} (- 2 (\frac{1}{12} ln 36 x^{2} + 12 x - 24 + \frac{ln \frac{6}{5} + \frac{6}{5} x - ln - \frac{4}{5} + \frac{6}{5} x}{60}) + 2 (\frac{1}{5} ln ∣ 3 x - 2 ∣ - \frac{1}{5} ln ∣ x + 1 ∣))

= 45 (\frac{1}{3} x + \frac{1}{3} (- 2 (\frac{1}{12} ln 36 x^{2} + 12 x - 24 + \frac{ln \frac{6}{5} + \frac{6}{5} x - ln - \frac{4}{5} + \frac{6}{5} x}{60}) + 2 (\frac{1}{5} ln ∣ 3 x - 2 ∣ - \frac{1}{5} ln ∣ x + 1 ∣)))

Add a constant to the solution

= 45 (\frac{1}{3} x + \frac{1}{3} (- 2 (\frac{1}{12} ln 36 x^{2} + 12 x - 24 + \frac{ln \frac{6}{5} + \frac{6}{5} x - ln - \frac{4}{5} + \frac{6}{5} x}{60}) + 2 (\frac{1}{5} ln ∣ 3 x - 2 ∣ - \frac{1}{5} ln ∣ x + 1 ∣))) + C

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What is the integral of (45x^2)/(3x^2+x-2) ?
The integral of (45x^2)/(3x^2+x-2) is