I have the integral
$$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$
which I am trying to integrate using complex integration. I know that each of the three terms has a fractional exponent, so I'd expect I need multiple branch cuts when I'm defining my contour. I have easily found the singularities (note $a\in [0,\infty)$ here). Most examples I've found online demonstrate how to solve for a branch cut with $\sqrt{x}$ but nothing like this.
Here's my attempt: I think I have seven branch points (not including those at infinity), since when the denominator is factored I end up with a total of six terms in the denominator and one term in the numerator, each with a fractional exponent. One approach would be to draw the branch cut from each branch point to infinity, radially outward. Another approach would be to draw a branch cut from the origin to each of the other six branch points, but I'm not sure where to go from there.