We want to find constants $A$, $B$, and $C$ such that
\[\frac{x+1}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1}.\]Multiplying both sides by $x^2(x-1)$, we get
\[x+1 = A x(x-1) + B(x-1) + Cx^2.\]Setting $x=0$, we get $B=-1$. Setting $x=1$, we get
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