The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite-element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law and ratio of shear to area dilatation moduli) on the capsule behaviour is investigated. Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary numbers, the capsule tumbles, behaving mostly like a solid particle. At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.