The procedure ellipsoid returns the surface area of an ellipsoid whose semi-axes are a, b, and c where $b\le a$, $c\le b$, $0\le c$. For a general ellipsoid the result is expressed as an unevaluated integral in terms of Int, which may be numerically evaluated by evalf.

When two or more of the arguments are the same, then a solvable integral can be created.  Otherwise, elliptical integrals are needed to evaluate the general case.

$\mathrm{evalf}\left(\mathrm{ellipsoid}\left(5\,3\,2\right)\right)$

$134.7751767$