I was trying to solve this integral problem and I noticed something that may be true
$$ \int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx = \frac{1}{2 n+1}\sum _{j=1}^{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, _2F_1\left(\frac{j}{r},-\frac{-j+2 n+1}{r};\frac{j}{r}+1;x^r\right) $$ for all $r>0, n\ge 1, n \in \mathbb Z$.
I did some numeric verification and this seems to be right. Any idea how to prove this?