How to solve this integral
$$ \int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r $$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in \mathbb{N} \right \} $ are some parameters.
This integral appears when computing the radial wave function normalization of a three-dimensional zero-range model in quantum mechanics, so I'm sure it converges.
For integer $\nu$ the hypergeometric function is a polynomial and one has $$\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-n,\tfrac{3}{2},\omega r^2)^2 \mathrm{d}r=\omega^{-3/2}\sqrt{\pi}\frac{(2 n+1)!}{2^{2 n+2}},\;\;n\in\mathbb{N}.$$
For noninteger $\nu$, Mathematica evaluates the integral $$\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,p,\omega r^2)^2 \mathrm{d}r$$ $$\qquad=\frac{\pi 4^{3 p-5} \Gamma (7-4 p) \Gamma (p-1) \, _3F_2\left(\frac{7}{2}-2 p,\frac{5}{2}-p,-\nu-p+1;-\nu-2 p+\frac{7}{2},2-p;1\right)}{\omega^{3/2}\Gamma (-\nu) \Gamma (2-p) \Gamma \left(-\nu-2 p+\frac{7}{2}\right)}$$ $$\qquad+\frac{\sqrt{\pi } 2^{-2 \nu-3} (3-2 p) \Gamma (2-2 p) \, _3F_2\left(\frac{3}{2},-\nu,\frac{5}{2}-p;-\nu-p+\frac{5}{2},p;1\right)}{\omega^{3/2}\left(-\nu-p+\frac{3}{2}\right) \Gamma (-2 \nu-2 p+2)}.$$ The hypergeometric functions are singular for $p=3/2$, numerically the limit $p\rightarrow 3/2$ agrees with a numerical evaluation of the integral, but I have not succeeded in obtaining a closed form expression for this limit.
