I wonder if Mathematica allows to compute the bivariate Meijer G function which is defined by means of a two-fold contour integral in the complex plane .
The integral of the product of three Meijer G functions can be expressed in closed form by means of the bivariate Meijer G function. Many useful integrals can be computed in this way.
 Agarwal, R.P.: An extension of Meijer's G-function. Proc. Nat. Inst. Sci. India, Part A 31,. 536- 546 (1965)