The hybrid expanding box (HEB) is used to study a slow compression of the magnetosheath plasma. The HEB simulations confirm the previous results of Hellinger et al. (2003b) that the slowly compressed plasma follows a marginal stability path of the proton cyclotron instability in a low-beta plasma, whereas in a high-beta plasma the mirror instability becomes dominant. For a faster compression the marginal stability path shifts further inside the unstable region in order to balance the faster anisotropization rate due to the faster compression. The presence of alpha particles strongly changes the dispersive properties of the plasma and introduces the alpha cyclotron instability. The theoretical description of the system becomes intricate, since the alpha particles introduce additional parameters. The plasma evolution during the compression gets more complex; a transition between the proton and the alpha cyclotron instabilities appears. The overall behavior of the simulated system is not qualitatively different from the behavior of the pure electron-proton plasma. When an additional parameter, a nonzero alpha/proton relative velocity, is included, the theoretical understanding becomes even more complicated. The behavior of the compressed system is nevertheless similar to the results in the pure electron-proton plasma. An important qualitative change from the zero alpha/proton relative velocity case is that the proton cyclotron instability is able to efficiently accelerate alpha particles with respect to protons, whereas the alpha cyclotron and the mirror instabilities decelerate them.