Any recommendations to help me to solve this integral $$ \int_{-\sqrt{x}+y}^{1-y} \left(1-(t+y)^2\right)^{r} \left(1-x(t-y)^2 \right)^{r} dt $$ where $\{r,x,y\} \in \mathbb{R} , \, |y|<1+x, \, \, r\ge0$, I used the taylor series to expand the $ \left(1-(t-y)^2\right)^{r}$ around $y=0$ and $\left(1-x(t-y)^2 \right)^{r}$ around $\sqrt{x}y=0$ and then i multiplied the expansions together.
After that i divided the resultant seris to 4 series, 1st and 2nd represents even-even and odd-odd terms and 3rd and 4th represents odd-even and even-odd terms. Now I am trying to solve these integrals:
1st $$ \int_{-x+y}^{1-y} (2t)^{2 (p+q)} \left(1-t^2\right)^{-m-p+r} \left(1-x t^2 \right)^{-n-q+r} dt $$ 2nd $$ \int_{-x+y}^{1-y} (2t)^{2 (p+q+1)} \left(1-t^2\right)^{-1-m-p+r} \left(1-x t^2 \right)^{-1-n-q+r} dt $$ 3rd $$ \int_{-x+y}^{1-y} (2t)^{2 (p+q)+1} \left(1-t^2\right)^{-1-m-p+r} \left(1-x t^2 \right)^{-n-q+r} dt $$ 4th $$ \int_{-x+y}^{1-y} (2t)^{2 (p+q)+1} \left(1-t^2\right)^{-m-p+r} \left(1-x t^2 \right)^{-1-n-q+r} dt $$ where $ \{m,n,p,q\} \in \mathbb{N}$. I think the solution will be in terms of hypergeometric functions or incomplete beta function, but i can't find the solution myself. I would be grateful if anyone can help.
All the antiderivatives are $$I(a,b,c)=\int t^a\, \left(1-t^2\right)^b \,\left(1-t^2 x\right)^c\,dt$$ $$I(a,b,c)=\frac{t^{a+1} }{a+1}\,F_1\left(\frac{a+1}{2};-b,-c;\frac{a+3}{2};t^2,t^2 x\right)$$ where appears the Appell hypergeometric function of two variables.