Integral of simple algebraic function

Question

I want to find an analytic expression for the following integral $$\int_{x_+}^Xdx\left(\frac{a}{x-b}+\frac{a^*}{x-b^*}\right)\frac{1}{\sqrt{(x-x_-)(x-x_+)}}\,,\quad a,b\in\mathbb{C}\,,\quad x_\pm\in\mathbb{R}\,,\quad x_+>x_-\,.$$ I know that an analytic expression in terms of elementary functions exists because Mathematica can give me one. The problem is that it is hideously and unnecessarily complicated, hence I want to evaluate it by hand. There is a formula for $a,b\in\mathbb{R}$ in Gradshteyn and Ryzhik, but not for complex $a,b$. Any help is greatly appreciated.

Answer 1

For $$\int \frac {dx}{(x-a)\sqrt{(x-b)(x-c)}}$$ $$x=\frac{c (a-b)+b (a-c) t^2} {(a-b)+ (a-c)t^2 }\implies dx=\frac{2 (a-b) (a-c) (b-c)\,t}{\big[(a-b)+ (a-c)t^2\big]^2}\,dt$$ makes also the antiderivative more than simple.