Control theory nerd here. I do a lot of work with differential equations, but I usually don't solve them exactly. I usually do other analyses like stability, controllability, observability, etc., where you don't need the closed-form solution to do your work. But I do know how to get numerical solutions.
Answer: depends:
Do you need to solve it? If you need to solve it, do you need an exact solution, or does a numerical approximation work? Do you have any initial or boundary conditions? Does this equation come from any physical system or model? What function space is f drawn from?
For proving existence and uniqueness (e.g. checking the conditions of Picard-Lindelöf) and deriving a numerical solution, my first step would be to convert it to the equivalent system x1'(t) = x2(t) and x2'(t) = ( x1(t)^-4^ ) f(t). (Here, you basically just add an extra state variable.) (We also get X(t) ≠ 0 unless f(t)=0.) Then, once I've checked that my initial conditions and driving function f will indeed produce a unique solution, use any numerical quadrature package to see what it looks like for different f and initial conditions, plugging in the right-hand sides of x1'(t) = x2(t) and x2'(t) = x1(t)^-4 f(t).
I could not personally derive a closed-form general solution of it as an initial value problem. I checked EqWorld's section on second order nonlinear ODEs and systems of two first-order ODEs, as well as my copy of Polyanin's handbook of exact solutions (LibGen) and I could not personally find a useful closed form solution. That doesn't mean it's not possibly in there, just that I'm not smart enough to find it.
I'm curious, how did you "fall upon" this differential equation?