My problem consists in evaluating $$\int_{y-\delta}^{x}dz \frac{1}{z^2\sqrt{a-b(z-y)^2}}$$ where $a,b>0$ and $y-\delta$ is one of the roots of $$a-b(z-y)^2=0$$ and both $y$ and $\delta$ are positive real numbers with $\delta < y$, so the integration is along a strictly positive domain. The end-point of the integration, $x$, should lie between $y-\delta$ and $y+\delta$.
What I've done so far is to plug this in Mathematica in the hope of obtaining a closed-form expression and I do obtain it, but it does not seem to be real (and it should).
Another idea that I had was to use complex integration, but I'm lacking some skills here... especially when it comes to branch cuts. Anyway, what I've managed was to but the integral in the form $$\int_{y-\delta}^{x}dz \frac{1}{z^2\sqrt{-b(z-z_{1})(z-z_{2})}}$$ where $z_{1,2}=y\pm \delta$.
I would really appreciate any help on this. Of course, I'm open to any method of obtaining a closed-form result, not necessarily complex integration. As a guideline for the numbers (if they would be needed): $y=10, a=20, b=3.8$ are very close to my actual values.