A generalized nonlinear Schrödinger equation is studied with the interplay between Kerr nonlinearity and intensity-dependent dispersion. The supported soliton solutions are characterized analytically in different families by the pseudo-potential method, in terms of Maimistov and Cuspon solitons for different ratio between the intensity-dependent dispersion and Kerr nonlinearity. Direct numerical simulations also agree with our analytical formulas. In addition to the well-studied Kerr-type nonlinearity, our results reveal an unexplored scenario with the introduction of the nonlinear corrections to wave dispersion.