$\int_0^1 (x^2-x^3)^{-1/3}\,dx$ by a dogbone contour

Compute \(\displaystyle\int_0^1 \frac{1}{\left(x^2 - x^3\right)^{1/3}}\) (hint: use the contour in figure 4);

Figure 4 of the source (integration contour for exercise 22) shows a dogbone contour around the segment \([0, 1]\): a segment just above \([0, 1]\) traversed from \(0\) to \(1\), a small circle around \(1\), a segment just below \([0, 1]\) traversed from \(1\) to \(0\), and a small circle around \(0\); this dogbone is enclosed by a large circle traversed counterclockwise.