Integrate Jacobi polynomials $w_{a+\epsilon,b}(x)P^{a,b}_k(x)P^{a,b}_l(x)$ with slightly modified weight

Question

It is well-known that the Jacobi polynomials satisfy the orthogonality relation $$\int^1_{-1}w_{ab}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx=\frac{2^{a+b+1}\Gamma(k+a+1)\Gamma(k+b+1)}{(2k+a+b+1)\Gamma(k+a+b+1)k!}\delta_{kl}\,,$$ where the weight function is given by $w_{ab}=(1-x)^a(1-x)^b$. I am trying to get evaluate the same integral with slightly modified weight, namely $$ I_1(\epsilon)=\int^1_{-1}w_{a+\epsilon,b}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx=\int^1_{-1}(1-x)^\epsilon w_{a,b}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx\,,\\ I_2(\epsilon)=\int^1_{-1}w_{a,b+\epsilon}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx=\int^1_{-1}(1+x)^\epsilon w_{a,b}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx\,, $$ i.e., the weights are just slightly modified. In the end, I would like to take the derivative with respect to $\epsilon$ at $\epsilon=1$. I can imagine that this calculation gets cumbersome, but I don't even quite know where to start.