Integration by parts or using u-sub? Integrate: $4(x^3+1)^{\frac{1}{3}}(x^5)$
I tried using u-sub but it is getting me absolutely nowhere. I also tried using integration by parts and eventually was able to evaluate the integral but it does not seem to simplify to the answer. Thank you for your help.

Maybe this will help: let $u = x^{3} + 1$.  Then $u - 1 = x^{3}$, and $du = 3x^{2}\, dx$ (i.e., $\frac{1}{3} \, du = x^{2} \, dx$), so we get:
$\int 4(x^{3} + 1)^{\frac{1}{3}}(x^{3}x^{2}) \,dx = \int \frac{4}{3}(u)^{\frac{1}{3}}(u - 1) \,du = \frac{4}{3} \int u^{\frac{4}{3}} - u^{\frac{1}{3}} \,du  = \frac{4}{3}(\frac{3}{7}u^{\frac{7}{3}} - \frac{3}{4}u^{\frac{4}{3}}) + C$
$ = \frac{4}{7}u^{\frac{7}{3}} - u^{\frac{4}{3}} + C$.
