We present the implementation of a Born-Oppenheimer (BO) hybrid quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) strategy using density functional theory (DFT) and the polarizable AMOEBA force field. This approach couples the Gaussian and Tinker suite of programs through a variational formalism allowing for a full self-consistent relaxation of both the AMOEBA induced dipoles and the DFT electron density at each MD step. As the DFT SCF cycles are the limiting factor in terms of computational efforts and MD stability, we focus on the latter aspect and compare the time-reversible BO (TR-BO) and the extended BO Lagrangian approaches (XL-BO) to the MD propagation. The XL-BO approach allows for stable, energy-conserving trajectories offering various perspectives for hybrid simulations using polarizable force fields.