These days, researchers are designing adaptive controllers for a range of applications because of their improved robustness and performance. Nevertheless, for under-actuated inverted pendulums, traditional adaptive controllers have not demonstrated performance that is sufficient. Therefore, in order to solve the aforementioned issue, a unique optimal fractional-order Massachusetts Institute of Technology (MIT) and Lyapunov rule are created for direct model reference adaptive control (MRAC) scheme in this work. To avoid time-domain analysis, an indirect method for higher-order continuing fraction expansions approximation is used to solve the fractional portion. Using a multi-objective function, the gray wolf optimizer (GWO) determines the changeable parameters of the aforementioned MRAC schemes (fractional-order parameter and adaptive gain). In simulation, comparisons are conducted between the FOPID controller, integer-order Lyapunov (IOLY)-based MRAC, fractional-order Lyapunov (FOLY), FOMIT rule-based MRAC, and integer-order MIT (IOMIT) rule-based MRAC schemes. A comparison with previous research is also displayed. To demonstrate the robustness of the proposed control technique, the effect of perturbing the plant model is examined. To demonstrate the effectiveness of the proposed approach, the effects of measurement noise and nonlinearity in plant dynamics are further investigated. The optimal FOLY approach that has been proposed performs better than the other schemes, as confirmed by the results.