Recently I've been studying on partial fractions and integration using partial fraction decomposition. I've not had any problems solving those types of integrals until I came across this integral:
$$ \int \left(\sqrt[6]{\dfrac{x}{x-2}} - \sqrt[4]{\dfrac{x}{x-2}}\right)\frac{\mathrm dx}{x^2-2x}$$
The book hints that you should substitute $\left( \dfrac{x}{x-2}=t^{12}, t \in \Bbb R \right)$. I've tried countless times but haven't found any way as to even end up with an integrand that can be decomposed into partial fractions.