Integral $\int^\infty_{-\infty}\int^\infty_{-\infty}(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1)^{-a_1-1}(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1)^{-a_2-1}dxdy$

Question

Under $x_i,y_i\in\mathbb R$, $s_i>0$ and $a_i>0$ for $i=1,2$, is there any good function to express the following integral? $$\int^\infty_{-\infty}\int^\infty_{-\infty} \left(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1\right)^{-a_1-1} \left(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1\right)^{-a_2-1}dxdy$$