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

$\begin{array}{rcl}I& =& \int \frac{\mathrm{dx}}{{(\mathrm{x}+1)}^{3/4}{(\mathrm{x}-2)}^{5/4}}\\ & =& \int \frac{\mathrm{dx}}{{\left({\displaystyle \frac{\mathrm{x}+1}{\mathrm{x}-2}}\right)}^{3/4}{(\mathrm{x}-2)}^2}\mathrm{dx}\\ \mathrm{Put} \frac{\mathrm{x}+1}{\mathrm{x}-2}& =& \mathrm{t} \Rightarrow \frac{-3}{{(\mathrm{x}-2)}^2}\mathrm{dx}=\mathrm{dt}\\ I& =& \frac{-1}{3}\int \frac{\mathrm{dt}}{{\mathrm{t}}^{3/4}}\\ & =& \left(\frac{-1}{3}\right)\frac{{\mathrm{t}}^{1/4}}{1/4}=\frac{-4}{3}{\left(\frac{\mathrm{x}+1}{\mathrm{x}-2}\right)}^{1/4}+\mathrm{c}\end{array}$