I am still sharpening my integration skills and I noticed that when the integrand is involving radicals and some power of $x$ such as $$\int x^5 \sqrt{x^3-2}dx,$$ I can let $\alpha=\sqrt{x^3-2}$ such that $\alpha^2+2=x^3$ and get $\frac 23 \alpha\, d\alpha = x^2dx$ which creates a neat substitution into $\alpha$ terms that makes the radical vanish and I get a result right away.
$$\begin{align} \int x^5 \sqrt{x^3-2}dx & = \int \left(x^3\right)\left(\sqrt{x^3-2}\right)\left(x^2dx\right) \\ & = \int \left(\alpha^2+2\right)\left(\alpha\right)\left(\frac 23 \alpha\,d\alpha\right) \\ & = \frac 23 \int \alpha^4+2\alpha^2\,d\alpha \\ & = \frac 2{15}\alpha^5+\frac 49 \alpha^3 \\ & = \frac 2{15}\left(x^3-2\right)^{5/2}+\frac 49 \left(x^3-2\right)^{3/2}+C \end{align}$$
I am not sure about this substitution, is it correct? I am not trying by any means to avoid doing the work of using integration by parts or other substitution techniques here, but rather I wonder if what I am doing with the radical and the differential in my substitution is allowed.
Thanks for the help!