In this work the scattering process of a massive vortex-antivortex dipole from a disklike obstacle has been investigated. The term “massive” refers to the fact that vortices of one species host the atoms of the other species, which thus play the role of massive cores. The motion of massless vortices is described by first-order motion equations, while the inclusion of core mass introduces a second-order time derivative that doubles the number of independent dynamical variables needed and is responsible for a much richer phenomenology. Infact, while complete integrability is reached in the massless case due to the presence of a sufficient number of conserved quantities to ensure such character, the introduction of core masses makes the system not-integrable as the number of degrees of freedom overtakes the number of conserved quantities.