The topic of dispersive optical solitons is one of the essential areas of telecommunications engineering. This paper presents the nonlinear Schrödinger equation (NLSE) with the cubic-quintic-septic (CQS) law which includes higher-order dispersion terms. This equation examines the dynamics of dispersive optical solitons propagation in nonlinear optical fibers. The presented model is significant in mathematics and theoretical physics sciences because long-distance optical fibers are used in optical communication utilized for transoceanic and intercontinental data transmission. First, the considered equation is transformed into a nonlinear ordinary differential equation (NLODE) using the traveling wave transformation. Then, two powerful techniques, Sinh-Gordon expansion approach and the auxiliary equation method (AEM), based on the homogeneous balance rule, are operated to produce analytical solutions for the NLSE with CQS law. Thus, bright, dark, singular, and periodic-singular soliton solutions are acquired. We demonstrate the three-dimensional (3D) and two-dimensional (2D) representations to interpret the physical behavior of some derived solutions. Besides, we investigate the effects of some parameters in the model with graphical illustrations. Therefore, it can be noted that the obtained soliton solutions have a significant role in soliton dynamics.