A certain multiple integral occurring in the studies of Beherens-Fisher multivariate problem has been evaluated by Mathai et al. (1995) in terms of invariant polynomials. However, this paper explicitly evaluates the context integral in terms of zonal polynomials, thus establishing a relationship between zonal polynomial integrals and invariant polynomial integrals.
A. K. Gupta. D. G. Kabe. "On a zonal polynomial integral." J. Appl. Math. 2003 (11) 569 - 573, 2 December 2003. https://doi.org/10.1155/S1110757X03209074