I'm trying to compute a closed form expression for the integral $ \int_{0}^{1}\int_{0}^{1}\frac{r^{i+j}(1-r)^{k+l}s^{2m-i-j}(1-s)^{2m-k-l}}{(r+s)^{m}(2-r-s)^{m}}drds \quad i,j,k,l \in\{0,1,\ldots,m\},$ which occurs in calculating $\mathrm{Ex}\left[\frac{R^{i}(1-R)^{k}S^{m-i}(1-S)^{m-k}}{(R+S)^{m}(2-R-S)^{m}}\right]$ where $R$ and $S$ are independent random variables with $R\sim Beta(j+1,l+1)$, $S\sim Beta(m-j+1,m-l+1)$ and $i,j,k,l \in\{0,1,\ldots,m\}$.
Right now the only idea I have is to try to find a partial fraction decomposition that will allow me to compute the integral of the resulting terms using integration by parts. However, this would be a lengthy and tedious calculation. I would really appreciate any ideas or suggestions for computing this integral in a more elegant way.