Can someone give me a hint on how to solve this problem?

Question

Problem with indefinite integral, can someone give a hint to do right substitute?

$$ \int \left(\frac{x^{\frac{1}{3}}-1}{x}\right)^{\frac{5}{3}}dx$$

Answer 1

Try $x=t^3$, then $t-1 = y^3$ and use Ostrogradsky approach afterwards.

Answer 2

By setting $x=y^3$ we have $dx=3y^2dy$ and $$ \int\left(\frac{x^{1/3}-1}{x}\right)^{5/3}\,dx = 3\int \frac{(y-1)^{5/3}}{y^3}\,dy $$ By setting $y=1+z^3$ the last integral turns into: $$ 9 \int \frac{z^7}{(1+z^3)^3}\,dz $$ and by partial fraction decomposition (yes, that is tedious but algorithmic) the last integral equals

$$\small \frac{1}{6} \left[\frac{9 z^2}{\left(1+z^3\right)^2}-\frac{24 z^2}{1+z^3}+10 \sqrt{3}\arctan\left(\frac{2z-1}{\sqrt{3}}\right)-10\log(z+1)+5\log\left(1-z+z^2\right)\right] $$