The noncentral Wishart density depends on a definite integral over the group of orthogonal matrices. This integral defines a function of the latent roots of a matrix involving the parent normal vectors and their means and covariances. An expansion for the integral in increasing powers of the reciprocals of these roots is developed using two distinct methods--an integration procedure and a substitution into a set of differential equations.
"An Asymptotic Expansion for the Noncentral Wishart Distribution." Ann. Math. Statist. 41 (5) 1700 - 1707, October, 1970. https://doi.org/10.1214/aoms/1177696814