Abstract: The equation of the fractionally nonlinear oscillator with strong quadratic damping force was considered with generalized damping term to Caputo fractional derivatives. The order of the derivatives considered for this problem was 0≤v≤1. At the lower end v = 0 the linearly damped harmonic oscillator and at the upper end v = 1 a non-linearly damped harmonic oscillator or strong quadratic damping force were obtained. The method of Laplace transformations were used to obtain the analytical solution. Eighteen roots were obtained against the usual three for ordinary (i.e., damped, over damped and critically damped). Two solutions were obtained for both positive and negative δ. For six of these cases it was shown that the frequency of oscillation increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative damping term.