How to evaluate $\int\frac{x^5}{x^3-2x^2-5x+6} dx$? None of substitution, partial fraction, and integration by parts seems to work here.

First note that by long division, 
$$
\frac{x^5}{x^3 - 2x^2 - 5x + 6} = x^2 + 2x + \frac{243}{10(x-3)} - \frac{1}{6(x-1)} - \frac{32}{15(x+2)} + 9. 
$$
Now, you can integrate the sum term by term, factoring out the constants to get as your answer
$$
\int \frac{x^5}{x^3 - 2x^2 - 5x + 6} dx = \frac{x^3}{3} + x^2 + 9x - \frac{1}{6} \log|1-x| + \frac{243}{10}\log|x-3| - \frac{32}{15} \log|x + 2| + c,
$$
where $c$ is a constant. 
