I need to numerically evaluate the following integral $$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$
where $c\ge 0$, and $n,\alpha$ are integers. I have no problem in evaluating this directly with available numerical packages in cases where $\alpha$ and $n$ are small. However, at large $n$ or $\alpha$ (say larger than ~100), the integrand becomes highly oscillatory, and even worse, the values of the Laguerre polynomial itself are no longer reliable (according the documentation of the package I'm using).
So, is there a method to simplify the integrand to mitigate these problems?