Low-energy excitations in quantum fluids are most directly encountered by ions. In the superfluid phases of He3 the relevant elementary excitations are Bogoliubov quasiparticles, which undergo repeated scattering off an ion in the presence of a divergent density of states. We present a quantum-mechanical calculation of the resonant He3 quasiparticle-scattering-limited mobility for negative ions in the anisotropic bulk A3 (A phase) and P3 (polar phase) that is exact when the quasiparticles scatter elastically. We develop a numerical scheme to solve the singular equations for quasiparticle-ion scattering in the A and P phases. Both of these superfluid phases feature a uniaxially symmetric order parameter but distinct topology for the magnitude of the energy gap on the Fermi sphere, i.e., points versus lines of nodes. In particular, the perpetual orbital circulation of Cooper pairs in A3 results in a novel, purely quantum mechanical intrinsic Magnus effect, which is absent in the polar phase, where Cooper pairs possess no spontaneous orbital angular momentum. This is of interest also for transport properties of heavy-fermion superconductors. We discuss the He3 quasiparticle-ion cross sections, which allow one to account for the mobility data with essentially no free parameters. The calculated mobility thus facilitates an introduction of ion spectroscopy to extract useful information on fundamental properties of the superfluid state, such as the temperature dependence of the energy gap in A3.