Frobenius & Special Functions · Superchallenge problem

Rodrigues' formula gives the Legendre polynomials as $P_n(x) = \dfrac{1}{2^n\,n!}\,\dfrac{d^n}{dx^n}\left(x^2 - 1\right)^n$. Compute $\displaystyle\int_{-1}^{1}x^8\,P_6(x)\,dx$. Express your answer as a common fraction.