What is the integral of (3x^3+6x^2+17x-5)/(x^2+2x+5) ?

The integral of (3x^3+6x^2+17x-5)/(x^2+2x+5) is

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integral of (3x^3+6x^2+17x-5)/(x^2+2x+5)

\int \frac{3 x ^{3} + 6 x ^{2} + 17 x - 5}{x ^{2} + 2 x + 5} d x

- 3 x - \frac{3}{2} ln \frac{1}{4} (x + 1)^{2} + 1 - 20 arctan (\frac{1}{2} (x + 1)) + \frac{33}{2} arctan (\frac{1}{2} (x + 1)) + \frac{5}{2} ln x^{2} + 2 x + 5 + \frac{3}{2} (x + 1)^{2} - 9 + C

Solution steps

\int \frac{3 x ^{3} + 6 x ^{2} + 17 x - 5}{x ^{2} + 2 x + 5} d x

Expand

\frac{3 x ^{3} + 6 x ^{2} + 17 x - 5}{x ^{2} + 2 x + 5} : \frac{3 x ^{3}}{x ^{2} + 2 x + 5} + \frac{6 x ^{2}}{x ^{2} + 2 x + 5} + \frac{17 x}{x ^{2} + 2 x + 5} - \frac{5}{x ^{2} + 2 x + 5}

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{3 x ^{3}}{x ^{2} + 2 x + 5} d x + \int \frac{6 x ^{2}}{x ^{2} + 2 x + 5} d x + \int \frac{17 x}{x ^{2} + 2 x + 5} d x - \int \frac{5}{x ^{2} + 2 x + 5} d x

\int \frac{3 x ^{3}}{x ^{2} + 2 x + 5} d x = 3 (\frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{1}{2} (- ln \frac{( x + 1 ) ^{2}}{4} + 1 + 11 arctan (\frac{x + 1}{2})))

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

\int \frac{17 x}{x ^{2} + 2 x + 5} d x = 17 (\frac{1}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1)))

\int \frac{5}{x ^{2} + 2 x + 5} d x = \frac{5}{2} arctan (\frac{1}{2} (x + 1))

= 3 (\frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{1}{2} (- ln \frac{( x + 1 ) ^{2}}{4} + 1 + 11 arctan (\frac{x + 1}{2}))) + 6 (- ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1)) + x) + 17 (\frac{1}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))) - \frac{5}{2} arctan (\frac{1}{2} (x + 1))

Simplify

3 (\frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{1}{2} (- ln \frac{( x + 1 ) ^{2}}{4} + 1 + 11 arctan (\frac{x + 1}{2}))) + 6 (- ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1)) + x) + 17 (\frac{1}{2} ln x^{2} + 2 x + 5 - \frac{1}{2} arctan (\frac{1}{2} (x + 1))) - \frac{5}{2} arctan (\frac{1}{2} (x + 1)) : - 3 x - \frac{3}{2} ln \frac{1}{4} (x + 1)^{2} + 1 - 20 arctan (\frac{1}{2} (x + 1)) + \frac{33}{2} arctan (\frac{1}{2} (x + 1)) + \frac{5}{2} ln x^{2} + 2 x + 5 + \frac{3}{2} (x + 1)^{2} - 9

= - 3 x - \frac{3}{2} ln \frac{1}{4} (x + 1)^{2} + 1 - 20 arctan (\frac{1}{2} (x + 1)) + \frac{33}{2} arctan (\frac{1}{2} (x + 1)) + \frac{5}{2} ln x^{2} + 2 x + 5 + \frac{3}{2} (x + 1)^{2} - 9

Add a constant to the solution

= - 3 x - \frac{3}{2} ln \frac{1}{4} (x + 1)^{2} + 1 - 20 arctan (\frac{1}{2} (x + 1)) + \frac{33}{2} arctan (\frac{1}{2} (x + 1)) + \frac{5}{2} ln x^{2} + 2 x + 5 + \frac{3}{2} (x + 1)^{2} - 9 + C

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What is the integral of (3x^3+6x^2+17x-5)/(x^2+2x+5) ?
The integral of (3x^3+6x^2+17x-5)/(x^2+2x+5) is