Frobenius & Special Functions · Superchallenge problem

Let $P_n$ be the Legendre polynomials, the polynomial solutions of $\left((1 - x^2)y'\right)' + n(n+1)y = 0$, normalised by $P_n(1) = 1$ and satisfying $\displaystyle\int_{-1}^1P_n^2\,dx = \frac{2}{2n+1}$. Compute $\displaystyle\int_{-1}^{1}(1 - x^2)\left[P_3'(x) + P_4'(x)\right]^2dx$. Express your answer as a common fraction.