We present the results of an analysis of the composition and energy spectra of cosmic rays for energies between $10$ TeV and $1$ PeV. For the study, we made use of a large collection of air shower data detected with HAWC during five years of observation. The corresponding measurements on the lateral shower age and the primary energy of the events were analyzed using the Gold unfolding algorithm, which allowed us to separate the elemental spectra for three mass groups: H, He and heavy nuclei ($Z > 2$). For the reconstruction of the spectra, we used CORSIKA/QGSJET-II-04 simulations. The results confirm the presence of softenings at tens of TeV in the spectra of the light mass groups of cosmic rays and hardenings at around $100$ TeV in these spectra. In addition, they also show an increment in the relative composition of heavy elements at high energies.