Kozlov and Onishchenko (1982) provide a condition under which the motion of an ellipsoid in inviscid fluid is integrable. We investigate the propensity for chaos in (a) triaxial ellipsoids and (b) ellipsoids of rotation in inviscid and viscous fluid. In particular we find a new integral of motion for an ellipsoid of rotation, and show how added mass is the main driver of the dynamics.