$\int \frac{1}{\sqrt[4]{{(x-1)}^3{(x+2)}^5}}dx=$

$I=\int \frac{1}{\sqrt[4]{{(x-1)}^3{(x+2)}^5}}dx$

$={\int }^{\text{}}\frac{1}{\sqrt[4]{{\left(\frac{x-1}{x+2}\right)}^3{(x+2)}^8}}dx$

$={\int }^{\text{}}\frac{1}{{(x+2)}^2}.\frac{1}{\sqrt[4]{{\left(\frac{x-1}{x+2}\right)}^3}}$

put $\frac{x-1}{x+2}=t\Rightarrow 1-\frac{3}{x+2}=t\Rightarrow \frac{3}{{(x+2)}^2}dx=dt$

$I=\int \frac{1}{3}\frac{dt}{t^{3/4}}$

$=\frac{1}{3}\int t^{-3/4}dt$

$=\frac{1}{3}\frac{t^{1/4}}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}+c$

$=\frac{4}{3}{\left(\frac{{\displaystyle x-1}}{{\displaystyle x+2}}\right)}^{\frac{1}{4}}+c$