<s>[INST] How to integrate $\frac{1}{(x+1)(x+2)^2(x+3)^3}$? I tried to solve it with partial fraction decomposition but the expression becomes way too difficult to solve. I could only solve three of six(A-F) expressions of the partial fraction expansion.
 [/INST] The substitution
$$x=-\left( 3+\frac{2}{u} \right)$$
reduces the integral to
$$\begin{align}
  & =\frac{1}{8}\int{\frac{{{u}^{4}}}{\left( u+1 \right){{\left( u+2 \right)}^{2}}}du} \\ 
 & =\frac{1}{8}\int{u-5+\frac{17{{u}^{2}}+36u+20}{\left( u+1 \right){{\left( u+2 \right)}^{2}}}du} \\ 
 & =\frac{1}{8}\int{u-5+\frac{1}{u+1}+\frac{16}{u+2}-\frac{16}{{{\left( u+2 \right)}^{2}}}du} \\ 
\end{align}$$
and you can see that partial fraction decomposition becomes much easier. 
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