Indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$

Question

Find the indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$:

I guess the fay forward would be to find a suitable substitution, but I am struggling with that.

Answer 1

Hint Expand the numerator $p(x)$ to get a polynomial $a_n x^n + \cdots a_1 x + a_0$. Then, dividing by $x \sqrt[3]{x} = x^{4 / 3}$ gives a linear combination of powers $x^{k - 4 / 3}$, each of which is straightforward to integrate.

Answer 2

If you do $x=y^3$ and $\mathrm dx=3y^2\,\mathrm dy$, then you'll get a rational function. Can you take it from here?