Please compute this integral:
$$I = \int_{| z | = 1} \frac{z^2+1}{z\sqrt{z (z-a)(z-1/a)}} dz$$
where $\vert a \vert < 1$. I have tried to construct dogbone integral or make a large circle $C_R$ with $R \to \infty$ in order to calculate the residue value at infinite. However, I do not know how to proceed. My description might be confusing, and it does not show much work that I have done but this is because I am still very uncertain about branch cut and square root in the function.