Cf. Dunkl et De. Ramirez, COMPUTING HYPERELLIPTIC INTEGRALS FOR SURFACE MEASURE OF ELLIPSOIDS, ACM transactions on mathematical software, 20(4), 1994, pp. 413-426
An algorithm for computing a class of hyperelliptic integrals and for
determining the surface measure of ellipsoids is described. The algori
thm is used to construct an omnibus optimal-design criterion.