Frobenius & Special Functions · Superchallenge problem

Using the Legendre generating function $\dfrac{1}{\sqrt{1 - 2xt + t^2}} = \displaystyle\sum_{n\ge0}P_n(x)\,t^n$ (valid for $|x| \le 1$, $0 \le t < 1$), compute $\displaystyle\int_{-1}^{1}\frac{P_5(x)}{\sqrt{1 - x}}\,dx$.