I will present some recent results concerning the generalization of the Harish-ChandraItzykson-Zuber integral to the case where the group integrated over is not UpN Dq, but the subgroup UpNq bD. This case is relevant for the study of D-partite quantum systems. While the HCIZ integral can no longer be expressed in terms of eigenvalues of the source matrices, it can be explicitly computed using Weingarten calculus. This result allows us to formulate a criterion for the detection of entanglement in multipartite quantum systems, which I will discuss in some detail.[-]