What is the integral of (x^3+ax^2+bx+c)/(x^2+7x+10) ?

The integral of (x^3+ax^2+bx+c)/(x^2+7x+10) is

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integral of (x^3+ax^2+bx+c)/(x^2+7x+10)

\int \frac{x ^{3} + a x ^{2} + b x + c}{x ^{2} + 7 x + 10} d x

\frac{1}{24} (12 x^{2} - 168 x + 468 ln (4 ∣ x + 5 ∣ ∣ x + 2 ∣) + 532 ln (2 ∣ x + 5 ∣) - 532 ln (2 ∣ x + 2 ∣) - 735) + a (x + \frac{7}{2} + 4 (- \frac{7}{8} ln 4 x^{2} + 28 x + 40 - \frac{29}{24} ln \frac{10}{3} + \frac{2}{3} x + \frac{29}{24} ln \frac{4}{3} + \frac{2}{3} x)) + 2 b (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{7}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x)) + c (\frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣) + C

Solution steps

\int \frac{x ^{3} + a x ^{2} + b x + c}{x ^{2} + 7 x + 10} d x

Expand

\frac{x ^{3} + a x ^{2} + b x + c}{x ^{2} + 7 x + 10} : \frac{x ^{3}}{x ^{2} + 7 x + 10} + \frac{a x ^{2}}{x ^{2} + 7 x + 10} + \frac{b x}{x ^{2} + 7 x + 10} + \frac{c}{x ^{2} + 7 x + 10}

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} + 7 x + 10} d x + \int \frac{a x ^{2}}{x ^{2} + 7 x + 10} d x + \int \frac{b x}{x ^{2} + 7 x + 10} d x + \int \frac{c}{x ^{2} + 7 x + 10} d x

\int \frac{x ^{3}}{x ^{2} + 7 x + 10} d x = \frac{1}{24} (12 x^{2} - 168 x + 468 ln (4 ∣ x + 5 ∣ ∣ x + 2 ∣) + 532 ln (2 ∣ x + 5 ∣) - 532 ln (2 ∣ x + 2 ∣) - 735)

\int \frac{a x ^{2}}{x ^{2} + 7 x + 10} d x = a (x + \frac{7}{2} + 4 (- \frac{7}{8} ln 4 x^{2} + 28 x + 40 - \frac{29}{24} ln \frac{10}{3} + \frac{2}{3} x + \frac{29}{24} ln \frac{4}{3} + \frac{2}{3} x))

\int \frac{b x}{x ^{2} + 7 x + 10} d x = 2 b (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{7}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x))

\int \frac{c}{x ^{2} + 7 x + 10} d x = c (\frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣)

= \frac{1}{24} (12 x^{2} - 168 x + 468 ln (4 ∣ x + 5 ∣ ∣ x + 2 ∣) + 532 ln (2 ∣ x + 5 ∣) - 532 ln (2 ∣ x + 2 ∣) - 735) + a (x + \frac{7}{2} + 4 (- \frac{7}{8} ln 4 x^{2} + 28 x + 40 - \frac{29}{24} ln \frac{10}{3} + \frac{2}{3} x + \frac{29}{24} ln \frac{4}{3} + \frac{2}{3} x)) + 2 b (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{7}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x)) + c (\frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣)

Add a constant to the solution

= \frac{1}{24} (12 x^{2} - 168 x + 468 ln (4 ∣ x + 5 ∣ ∣ x + 2 ∣) + 532 ln (2 ∣ x + 5 ∣) - 532 ln (2 ∣ x + 2 ∣) - 735) + a (x + \frac{7}{2} + 4 (- \frac{7}{8} ln 4 x^{2} + 28 x + 40 - \frac{29}{24} ln \frac{10}{3} + \frac{2}{3} x + \frac{29}{24} ln \frac{4}{3} + \frac{2}{3} x)) + 2 b (\frac{1}{4} ln 4 x^{2} + 28 x + 40 + \frac{7}{12} (ln \frac{10}{3} + \frac{2}{3} x - ln \frac{4}{3} + \frac{2}{3} x)) + c (\frac{1}{3} ln ∣ x + 2 ∣ - \frac{1}{3} ln ∣ x + 5 ∣) + C

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What is the integral of (x^3+ax^2+bx+c)/(x^2+7x+10) ?
The integral of (x^3+ax^2+bx+c)/(x^2+7x+10) is