A Contour Integral I What is the value of the integral
\begin{align}
\int_{-a}^{c} \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)} 
\end{align} 

Hint
This integral simplifies if you first make a change of variable $$ \sqrt{ \frac{a+x}{c-x} }=u^2$$ that is to say $$x=\frac{c u^2-a}{u^2+1}$$ So $$I=\int \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)}=-\int \frac{2 u^2 (a+c)~~ du}{\left(u^2 (b-c)+a+b\right) \left(u^2 (d-c)+a+d\right)}$$ Using partial fractions, this leads to $$I=\int \frac{2 (a+b)}{(b-d) \left(u^2 (b-c)+a+b\right)}- \int \frac{2 (a+d)}{(b-d) \left(u^2 (d-c)+a+d\right)}$$ Using $u$ as variable, the bounds are $0$ and $\infty$.
I am sure that you can take from here.
