None of substitution, partial fraction, and integration by parts seems to work here.
2026-04-06 09:55:37.1775469337
How to evaluate $\int\frac{x^5}{x^3-2x^2-5x+6} dx$?
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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.