Consider u = f'(x), then:
du/dx + a*u^2 + b*u = 0 or
du/dx = -a*u^2 - b*u which is separable.
While harder for this problem, in general, any 2nd order ODE can be reduced to a first order ODE by the substitution
By the chain rule we get
This will change from the dependent variable to the independent variable of a new function u.
This can be solved by an integrating factor.