numerical integration – Is there a way to speed up Integrate when the integrand contains a product of polynomials each of which having a large degree?

I have integrals of the form
$$int_0^inftymathrm d x; e^{-x^2};;(textrm{Polynomial of degree n})times(textrm{Polynomial of degree m}),$$
where $n,m$ can be as large as 300. (If this is helpful, these polynomials are Laguerre polynomials, provided by the function LaguerreL(...)).

When I use Integrate(f(x), {x, 0, Infinity}), the calculation is too slow for large polynomial degrees.

Is there a way to speed up this calculation? (I tried to use NIntegrate, but unfortunately, since the integrand is highly oscillatory at large $n,m$, I’m unable to get reliable results from numerical methods).