I want to find the a 3 term perturbation soln of (i) $(1+x)^3 = ex$ where $e\ll1$

Direct substitution of the regular perturbation series $x = x_0 + ex_1 + e^2x_2$ into (i) does not work

I think soln has the form: $x = x_0 + e^{1/3}*x_1 + e^{2/3}*x_2$ Seems to work, but not sure it is correct

TIA, Matt