We studied the intensity of resonant Raman scattering due to optical phonons in a planar II-VI-type semiconductor microcavity in the regime of strong coupling between light and matter. Two different sets of independent experiments were performed at near outgoing resonance with the middle polariton (MP) branch of the cavity. In the first, the Stokes-shifted photons were kept at exact resonance with the MP, varying the photonic or excitonic character of the polariton. In the second, only the incoming light wavelength was varied, and the resonant profile of the inelastic scattered intensity was studied when the system was tuned out of the resonant condition. Taking some matrix elements as free parameters, both independent experiments are quantitatively described by a model which incorporates lifetime effects in both excitons and photons, and the coupling of the cavity photons to the electron-hole continuum. The model is solved using a Green's function approach which treats the exciton-photon coupling nonperturbatively. © 2008 The American Physical Society.