I am curious to know if a closed form expression can be given for the integral, \begin{equation} \int_0^\infty\, dx\, e^{-cx} (\sqrt x)^r J_s (b \sqrt{x}) L_m^\gamma(cx) L_n^\lambda(cx)\nonumber \end{equation}
Following cases are known,
1) $r = s = \gamma + \lambda$ [Koelbig et al, J. Comput. Appl. Math. 71 (1996) 357]. Gradshteyn or Prudinikov also tabulate this case albeit with wrong result.
2) $r = s$ [Shukla et al, Commun. Korean Math. Soc. 27 (2012), No. 4, 721]