Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Question

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} $$ for $0<a<b<c<d$? Here $\max(r,r') = r$ if $r\geq r'$ and $\max(r,r') = r'$ otherwise, $\min(r,r')$ is defined similarly. For large $\ell$, it may be shown that the integral is dominated by $r\approx r'$ and asymptotically $$ S_{\ell} \rightarrow \frac{2}{\ell}\int\limits_{b}^{c} dr \, \frac{r^2}{\sqrt{(r - a)(r - b)(r-c)(r-d)}} $$ This can be integrated in a closed form using elliptic integrals as shown here. That solution seemed so general, I wonder if the double integral can also be integrated similarly in a closed form.