The surface area of an $n$-dimensional ellipsoid is expressed in terms of hyperelliptic integrals, see SURFACE AREA AND CAPACITY OF ELLIPSOIDS IN n DIMENSIONS by Garry J. Tee
In dimension 3, they are elliptic integrals, and the result is due to Legendre. The paper also mentions approximate formulas.
For $n\neq m\neq p$ one can write the integrals but they have no standard name: Surface area of an $\ell_p$ unit ball?