What is the integral of (9x^2-25x-10)/(x^3-4x^2-5x) ?

The integral of (9x^2-25x-10)/(x^3-4x^2-5x) is 4ln|x+1|+3ln|x-5|+2ln|x|+C

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integral of (9x^2-25x-10)/(x^3-4x^2-5x)

\int \frac{9 x ^{2} - 25 x - 10}{x ^{3} - 4 x ^{2} - 5 x} d x

4 ln ∣ x + 1 ∣ + 3 ln ∣ x - 5 ∣ + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{9 x ^{2} - 25 x - 10}{x ^{3} - 4 x ^{2} - 5 x} d x

Expand

\frac{9 x ^{2} - 25 x - 10}{x ^{3} - 4 x ^{2} - 5 x} : \frac{9 x ^{2}}{x ^{3} - 4 x ^{2} - 5 x} - \frac{25 x}{x ^{3} - 4 x ^{2} - 5 x} - \frac{10}{x ^{3} - 4 x ^{2} - 5 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{9 x ^{2}}{x ^{3} - 4 x ^{2} - 5 x} d x - \int \frac{25 x}{x ^{3} - 4 x ^{2} - 5 x} d x - \int \frac{10}{x ^{3} - 4 x ^{2} - 5 x} d x

\int \frac{9 x ^{2}}{x ^{3} - 4 x ^{2} - 5 x} d x = 9 (\frac{1}{6} ln ∣ x + 1 ∣ + \frac{5}{6} ln ∣ x - 5 ∣)

\int \frac{25 x}{x ^{3} - 4 x ^{2} - 5 x} d x = 25 (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 5 ∣)

\int \frac{10}{x ^{3} - 4 x ^{2} - 5 x} d x = 10 (- \frac{1}{5} ln ∣ x ∣ + \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{30} ln ∣ x - 5 ∣)

= 9 (\frac{1}{6} ln ∣ x + 1 ∣ + \frac{5}{6} ln ∣ x - 5 ∣) - 25 (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 5 ∣) - 10 (- \frac{1}{5} ln ∣ x ∣ + \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{30} ln ∣ x - 5 ∣)

Simplify

9 (\frac{1}{6} ln ∣ x + 1 ∣ + \frac{5}{6} ln ∣ x - 5 ∣) - 25 (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x - 5 ∣) - 10 (- \frac{1}{5} ln ∣ x ∣ + \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{30} ln ∣ x - 5 ∣) : 4 ln ∣ x + 1 ∣ + 3 ln ∣ x - 5 ∣ + 2 ln ∣ x ∣

= 4 ln ∣ x + 1 ∣ + 3 ln ∣ x - 5 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= 4 ln ∣ x + 1 ∣ + 3 ln ∣ x - 5 ∣ + 2 ln ∣ x ∣ + C

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What is the integral of (9x^2-25x-10)/(x^3-4x^2-5x) ?
The integral of (9x^2-25x-10)/(x^3-4x^2-5x) is 4ln|x+1|+3ln|x-5|+2ln|x|+C