Double complex branch cut integration, how to?

Question

I'm working on a small project of mine, but I have gotten stuck on a particularly challenging integral: $$\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^2+1}\sqrt{x^2+ax+b}}$$ With the condition that $a^2-4b<0$, so all roots will be complex-valued. I'd like to try contour integration around the positive semicircle with center $0$, but I haven't been able to make it work. Even taking branch cuts as the half-lines $[i, \infty]$ and $[\frac{1}{2}\left(-a + i\sqrt{4b-a^2}\right), \infty]$ did not work, as the estimation lemma decides to render every integral around any point to $0$, and the integral over the semicircle itself goes to zero as well, as the denominator contains 2 factors of R. What has gone wrong?The contour I've chosen. For clarity I've chosen the nontrivial branch point in the first quadrant, but it might at well be in the second. I can't quite figure out where the nonzero value is hiding, as all integrals I evaluate end up being $0$