<s>[INST] Evaluating $\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$ $$\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$$
I am suppose to make a $u$ substitution and to make this a rational integral and then evaluate it from there but I have no idea how to do that. There aren't any good examples of this in the book and I can not find any us that make this doable. 
 [/INST] Let $u = x^2 + 1$, $du = 2x\,dx$:
\begin{align*}
\int \frac{x^3\,dx}{(x^2 + 1)^\frac{1}{3}} &= \int \frac{x^2\,x\,dx}{(x^2 + 1)^\frac{1}{3}} = \frac{1}{2} \int \frac{u-1}{u^\frac{1}{3}}\,du = \frac{1}{2} \int \left(u^\frac{2}{3} - u^{-\frac{1}{3}}\right) \,du \\
&= \frac{3}{10} u^{\frac{5}{3}} - \frac{3}{4} u^{\frac{2}{3}} + C \\
&= \frac{3}{5} \left(x^2+1\right)^{\frac{5}{3}} - \frac{3}{4} \left(x^2+1\right)^{\frac{2}{3}} + C
\end{align*}
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