How do I evaluate this integral: $ \int \frac{x^2+3x-2}{x^5+x^4+x^3-x^2-x-1}dx$

Question

How do I evaluate this integral: $ \int \frac{x^2+3x-2}{x^5+x^4+x^3-x^2-x-1}dx$

I already got it into this form $-\frac{2}{9}\int \frac{x}{x^2+x+1}dx-\frac{4}{9}\int\frac{1}{x^2+x+1}dx +\frac{1}{3}\int\frac{x}{(x^2+x+1)^2}dx+\frac{8}{3}\int\frac{1}{(x^2+x+1)^2}dx+\frac{2}{9}\int\frac{1}{x-1}dx$

(partical fraction decomposition)

But I cannot proceed :/

How can I deal with all the annoying quotients?

Answer 1

Use $\left( \frac{5x+4}{x^2+x+1}\right)’ =- \frac5{x^2+x+1} -\frac{3(x-2)}{(x^2+x+1)^2}$ to integrate in compact form below

\begin{align} & \int \frac{x^2+3x-2}{x^5+x^4+x^3-x^2-x-1}dx = \int \frac{x-2}{(x^2+x+1)^2}dx\\ = & -\frac13 \frac{5x+4}{x^2+x+1}- \frac53 \int \frac{1}{x^2+x+1}dx\\ = & -\frac13 \frac{5x+4}{x^2+x+1}- \frac{10}{3\sqrt3} \tan^{-1}\frac{2x+1}{\sqrt3}+C \end{align}