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Abstract This paper focuses on the numerical investigation of a strongly nonlinear oscillator with cubic and harmonic restoring force. We transform this oscillator as a free damped cubic-quintic Duffing oscillator equation by Taylor approximation. The approximated solutions with high accuracy are provided by using the global residue harmonic balance method (GRHBM) without any discretization or restrict assumptions. The sensitive analysis of the approximation or the frequency with respect to the amplitude is considered in detail. Numerical comparisons with Runge–Kutta method and harmonic balance method are given to show the efficiency and stability of GRHBM.