$\displaystyle \int \frac{x+2}{x^3(x-1)}dx$
$\displaystyle \begin{aligned}
\frac{x+2}{x^{3}(x-1)}&=\frac{x-1-3\left( x^{3}-1 \right)+3x^{3}}{x^{3}(x-1)} \\
& =\frac{(x-1)\left( 1-3x^{2}-3x-3 \right)+3x^{3}}{x^{3}(x-1)} \\
& =\frac{3}{x-1}-\frac{3x^{2}+3x+2}{x^{3}} \\
& =\frac{3}{x-1}-\frac{3}{x}-\frac{3}{x^{2}}-\frac{2}{x^{3}}.
\end{aligned}$
Integrate.