$$\oint_{|z|=1}\frac{dz}{\sqrt{(z-z_1)(z-z_2)(z-z_3)(z-z_4)}}$$ where $z_1$ and $z_4$ is in the contour but $z_2$ and $z_3$ is out of the contour. And the branch cut can be taken between $z_1$ and $z_4$ and between $z_2$ and $z_3$ in order to avoid the intersection of the contour and branch cuts.
PS: If there is no analytical expression for the above integral, how about this $$\oint_{|z|=1}\frac{dz}{\sqrt{(z-1+\delta)(z-1-\delta)(z+1+\delta)(z+1-\delta)}}$$ where $\delta>0$ and $\delta\rightarrow 0$