The rate constant for the H atom abstraction reaction from methane by the muonic helium atom, He mu + CH4 -> He mu H + CH3, is reported at 500 K and compared with theory, providing an important test of both the potential energy surface (PES) and reaction rate theory for the prototypical polyatomic CH5 reaction system. The theory used to characterize this reaction includes both variational transition-state (CVT/mu OMT) theory (VTST) and ring polymer molecular dynamics (RPMD) calculations on a recently developed PES, which are compared as well with earlier calculations On different PESs for the H, D, and Mu + CH4 reactions, the latter, in particular, providing for a variation in atomic mass by a factor of 36. Though rigorous quantum calculations have been carried out for the H + CH4 reaction, these have not yet been extended to the isotopologues of this reaction (in contrast to H-3), so it is important to provide tests of less rigorous theories in comparison with kinetic isotope effects measured by experiment. In this regard, the agreement between the VTST and RPMD calculations and experiment for the rate constant of the He mu + CH4 reaction at 500 K is excellent, within 10% in both cases, which overlaps with experimental error.