$\frac{d}{dx}\left(\frac{\left(x-2\right)^2}{\left(x+1\right)\left(x-3\right)}\right)=\frac{2\left(x-2\right)\left(x+1\right)\left(x-3\right)+\left(-x+3-x-1\right)\left(x-2\right)^2}{\left(x+1\right)^2\left(x-3\right)^2}$
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Integral
$\int\frac{\left(x-2\right)^2}{\left(x+1\right)\left(x-3\right)}dx=x+\frac{1}{4}\ln\left(x-3\right)-\frac{9}{4}\ln\left(x+1\right)+C_0$
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