Suppose I have the Hermite functions (including the exponential weight function) $\{\phi_m\}$ that form an orthonormal basis of $L^2(\mathbb{R})$. I consider the physicists version in https://en.wikipedia.org/wiki/Hermite_polynomials
My question is, is there a nice closed form expression for the integral
$$ \int_{0}^\infty \phi_m(x)\phi_n(x)\frac{1}{\sqrt{x}}dx $$
for $m,n$ odd? Preferably I would like an expression which doesn't involve too many cancelling factorials since I want to evaluate it on a computer for $m,n$ up to about $2000$.