In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials $L_n^\alpha(x)$. To perform the decomposition I have to compute the following definite integral

$$ T_n:=\int_0^\infty L_k^\alpha(x)^3 L_n^\alpha(x)x^{\alpha-\delta}e^{-x}dx\,. $$

Is there a closed form formula for the above integral or a way to obtain it?

EDIT:

I have found a paper by Artur Erdélyi where he gives a general formula for an integral of a product of $n$ Laguerre polynomials. Unfortunately, the result is presented in terms of Lauricella hypergeometric function and is not easily computable in the case of $T_n$. There are also some papers on a product of three Laguerre polynomials, which might be used to simplify $T_n$ but I won't be able to read them thoroughly for several days now.