Contour integral with square roots in numerator and denominator

Contour integral with square roots in numerator and denominator

trying to workout the following integral:
$$\oint_{\gamma}\frac{\sqrt{(x-a)(x-b)}}{\sqrt{x^2-1}} , \quad 1<a<b$$
where $\gamma$ represents dumbbell contour between $a$ and $b$. Thanks in
advance