This paper examines the cubic-quintic nonlinear Schrödinger equation (CQNLSE) with an additional anti-cubic nonlinear term, by using Stainberg's symmetry technique. The CQNLSE with the additional anti-cubic nonlinear term is a generalized model of higher-order nonlinear effects, offering a more accurate description of optical pulse propagation in nonlinear media with complex nonlinear responses, which makes the CQNLSE have a wide range of applications in several fields like optics, communications, spectroscopy, and computing. In our study, we used symmetry group analysis to derive a finite Lie group of transformations, and as a result, a novel similarity transformation, not previously reported in the literature, was obtained from this group. By using this transformation, the CQNLSE with the anti-cubic term was reduced to a nonlinear ordinary differential equation, which can be solved using the Jacobi elliptic expansion method, and a variety of wave solutions were obtained. These solutions include periodic waves, kink solitons, and bright solitons, contain other solutions, shown in the previous literature. We have also introduced a new solution, which has not been achived before in studies. The 3D and 2D plots of the periodic sn wave and its limit as a kink solitary wave were given to declare the dynamical behavior of the wave propagation by controlling the parameters contained in the solution.