While applying Cauchy's residue theorem, we often need to exclude branch points from the enclosed area of the contour.
Well, we still have to evaluate the small contour integration over the small loops around branch points.
Doing Laurent series expansion and then do term by term integration is not feasible as a Laurent series is not available at branch points.
I’ve seen some related examples: $$\oint\sqrt{z}\,\mathrm{d}z=\int_0^{2\pi}r^{\frac12}\,\mathrm{e}^{\frac{\mathrm{i}\theta}2}\,\mathrm{i}\,r\mathrm{e}^{\mathrm{i}\theta}\,\mathrm{d}\theta$$ which equals $$\frac{-4}3r^\frac32$$ and approaches zero as $r$ tends to zero.
However, when functions cannot be integrated easily, how can we compute such contour integral around branch points?
Moreover, are the cases different for algebraic branch points and logarithmic branch points?
An integral over a path that crosses a branch cut may depend on where you cross the branch cut (thus the $r^{3/2}$ in your example).
For algebraic branch points you might use a Puiseux series around the branch point.