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

The integral of (1+x+x^2)/((x^2+1)(2x+1)) is 1/5 (1/2 ln|x^2+1|+2arctan(x))+3/10 ln|2x+1|+C

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

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

\frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{3}{10} ln ∣ 2 x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{1 + x + x ^{2}}{( x ^{2} + 1 ) ( 2 x + 1 )} : \frac{x + 2}{5 ( x ^{2} + 1 )} + \frac{3}{5 ( 2 x + 1 )}

= \int \frac{x + 2}{5 ( x ^{2} + 1 )} + \frac{3}{5 ( 2 x + 1 )} 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

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

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

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

= \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{3}{10} ln ∣ 2 x + 1 ∣

Add a constant to the solution

= \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{3}{10} ln ∣ 2 x + 1 ∣ + C

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