I'm looking for some help integrating the following,
$$\int_a^b \frac{x}{\sqrt{(b-x)(x-a)}} \, J_0(kx) \, dx$$
I've found similar integrals in the tables of Gradstein and Ryzhik where the irrational term is $\sqrt{c-u^2}$. I can complete the square and $u$ sub but I end up with the argument of the Bessel function being a sum of $u$ and a constant, and I do not know how to proceed. Further, I am not sure if I can transform the integral limits to match the expression in the tables. I'm not confident that there is even a closed form expression for this integral.
Does anyone have any suggestions on how to integrate this not using tables? What is the standard method used for integrating functions like this?