This paper recovers pure-cubic soliton solutions when chromatic dispersion, being negligibly small, is discarded and replenished with third-order dispersion. The governing model is considered with a few Hamiltonian perturbation terms that will sustain the balance between dispersion and self-trapping effect stemming from the nonlinear refractive index. The model is studied with ten forms of nonlinearity. The unified Riccati equation expansion method is the integration algorithm utilized here. This yields dark and singular optical solitons that are all presented along with their respective parameter restrictions.