I have tried simplifying the integral (turning into an integral in terms of position variables) using the equation of motion. It's easy to show:
$$\frac{1}{x^4}=\frac{(\ddot{x})^2}{(GM)^2} $$
And therefore one can write:
$$\frac{1}{(GM)^2}\int_{t=0}^{t_f} (\dot{x}\ddot{x})^2 dt =...