For the first time, we find the complex solitons for a quasi-one-dimensional Bose–Einstein condensate with two- and three-body interactions. These localized solutions are characterized by a power law behaviour. Both dark and bright solitons can be excited in the experimentally allowed parameter domain, when two- and three-body interactions are, respectively, repulsive and attractive. The dark solitons travel with a constant speed, which is quite different from the Lieb mode, where profiles with different speeds, bounded above by sound velocity, can exist for specified interaction strengths. We also study the properties of these solitons in the presence of harmonic confinement with time-dependent nonlinearity and loss. The modulational instability and the Vakhitov–Kolokolov criterion of stability are also studied.