In this paper, we present the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms that define the propagation of ultrashort pulses. Two analytical methods, which are the new Kudryashov’s method and the unified Riccati equation expansion method, are implemented to extract the analytical soliton solutions of the presented equation for the first time. Thus, bright, dark, and singular soliton solutions are acquired. To illustrate the physical behavior of some of the obtained solutions, 3D, 2D, and contour graphs are depicted. In particular, to understand the effects of the group velocity dispersion, 3OD, 4OD, CQNLs, self-steepening coefficient terms, and group velocity term of the traveling wave transformation on the soliton dynamics of the proposed equation, 2D plots for different values of coefficients are represented. The obtained results provide us with the knowledge that the presented model can be examined from a physical perspective. It can be concluded that the used methods are effective approaches to derive the analytical solutions for the NLSE.