We know the surface area of a supersphere (x^n+y^n+z^n=1) can be easily obtained using the gamma function (here is the discussion). But what happens when we consider a superellipsoid ($x^m +y^n+z^p=1; m \neq n\neq p $), or more generally for the implicit function $(x/a)^m +(y/b)^n+(z/c)^p=1; a \neq b\neq c; m \neq n\neq p $
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?