Assuming that $a$ and $b$ are positive real numbers, and $c$, $d$, $x_{0}$, and $y_{0}$ are all real. I'm trying to find a closed-form expression for the following definite integral. \begin{equation} I = \int_{-a}^{a}\int_{-b}^{b} \frac{\sqrt{\left(x^{2}+c^{2}\right)\left((x-x_{0})^{2}+d^{2}\right)}}{\left(\left(x^2+y^2+c^2\right)\left((x-x_{0})^2+(y-y_{0})^{2}+d^2\right)\right)^{\tfrac{5}{4}}} dydx. \end{equation}
Do you know how I can derive the closed-form value of $I$? I'm also looking to find a closed-form expression of $I$ for $a\to\infty$ and $b\to\infty$.