The countless applications of nonlinear evolution equations (NLEEs) in diverse scientific fields have attracted the attention of many academic scholars to search and analyze their exact solutions. In this respect, various methods such as the Kudryashov method [1–7], tanh-coth method [8, 9], modified simple equation method [10, 11], sine-cosine method [12, 13], transformed rational function method [14, 15], ansatz method [16, 17], auxiliary equation method [18, 19], semi-inverse variational method [20, 21], and expa-function method [22–24] have been utilized to solve and handle the NLEEs. The expa-function scheme is one of the most practical techniques to acquire the exact solutions of NLEEs and is widely exerted to retrieve the exact solutions of NLEEs; for example, the combined KdV–mKdV equation [22], the unstable nonlinear Schrödinger equation [23], and the Tzitzéica-type equations [24]. In the present paper, the dispersive cubic-quintic Schrödinger equation (DCQSE) including higher-order time derivatives is studied through the expa-function scheme. The nonlinear governing model in its dimensionless form is presented as below [25–27]: iuz− α1