An integral of the form $$\frac{n_2!}{n_1!}\int_0^{\infty}\mathrm dx\,e^{-x}x^{n_1-n_2}{[L_{n_2}^{n_1-n_2}(x)]}^2f(x),$$ (where $n_1\ge n_2\ge 0$ are integers, $L$ is an associated Laguerre polynomial, and $f$ is a function of the form $f(x)=1/\sqrt{x+a}$, with $a\ge0$) can be approximated at large $n_2$ or large $n_1-n_2$ by the simpler integral $$\int_0^\pi\frac{\mathrm d\phi}{\pi}f\bigl(n_1+n_2-2\sqrt{n_1n_2}\cos\phi\bigr).$$
I found this approximation in Appendix B of this paper. However, I have no idea how this approximation was obtained. The mentioned paper doesn't point to any references when it uses this approximation. It mentions that it's called a "semiclassical approximation", but I don't know what that means, and searching online didn't help.
Can you help me derive this approximation? I don't know where to start.