The integral $\int \frac{1}{\sqrt[4]{{(x-1)}^3{(x+2)}^5}}dx$ is equal to 

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

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

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

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

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

$=\frac{1}{3}\frac{t^{-{\displaystyle \frac{3}{4}}+1}}{-\frac{3}{4}+1}+c$

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