Based on a continuous internal energy state variable, we propose an explicit, fully nonlinear Boltzmann collision operator for the evolution of the distribution function describing a polyatomic gas with a constant heat capacity. The particle interaction is a polyatomic generalization of the variable hard-sphere model, used in a recent rigorous mathematical analysis, and includes frozen collisions. The model is consistent with the monatomic case and allows easy evaluations for moment equations and the Chapman-Enskog expansion. Using a publicly available computer algebra code we can explicitly compute nonlinear production terms for macroscopic systems of moments. The range of Prandtl number values recovers the Eucken formula for a specific choice of frozen collisions.