Frobenius & Special Functions · Superchallenge problem

Using the Legendre generating function $\dfrac{1}{\sqrt{1 - 2xt + t^2}} = \displaystyle\sum_{n\ge0}P_n(x)t^n$ and the orthogonality of the $P_n$ on $[-1, 1]$, compute $\displaystyle\int_{-1}^{1}\frac{P_3(x)}{\sqrt{5/4 - x}}\,dx$. Express your answer as a common fraction.