The response of the ${{\mathrm{D}}_{3}}^{+}$ and ${{\mathrm{H}}_{3}}^{+}$ molecules to an intense, ultrashort laser pulse is investigated with a quasiclassical model to provide details of our previous short report [Phys. Rev. Lett. 106, 203001 (2011)]. The stable, lowest-energy configuration of two electrons and three deuterons (or three protons) is realized by certain auxiliary, momentum-dependent potentials. The time evolution of the molecule in the external driving field leading to the dissociation and ionization is calculated by numerically solving the classical Hamiltonian equations of motion. It is shown that the theoretical results reproduce qualitatively characteristic features of the experimental angular distributions and kinetic energy release spectra [Phys. Rev. Lett. 103, 103004 (2009)].