What is the integral of (x^3+2x^2+5x+1)/(x^2+2x+5) ?

The integral of (x^3+2x^2+5x+1)/(x^2+2x+5) is

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integral of (x^3+2x^2+5x+1)/(x^2+2x+5)

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

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

Solution steps

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

Expand

\frac{x ^{3} + 2 x ^{2} + 5 x + 1}{x ^{2} + 2 x + 5} : \frac{x ^{3}}{x ^{2} + 2 x + 5} + \frac{2 x ^{2}}{x ^{2} + 2 x + 5} + \frac{5 x}{x ^{2} + 2 x + 5} + \frac{1}{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{x ^{3}}{x ^{2} + 2 x + 5} d x + \int \frac{2 x ^{2}}{x ^{2} + 2 x + 5} d x + \int \frac{5 x}{x ^{2} + 2 x + 5} d x + \int \frac{1}{x ^{2} + 2 x + 5} d x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of (x^3+2x^2+5x+1)/(x^2+2x+5) ?
The integral of (x^3+2x^2+5x+1)/(x^2+2x+5) is