Necessary time to let a particle at equilibrium point

I remember that there exists a demonstration showing that this time is actually infinite, but I cannot find it. We have a potential $V(x)$ that has a local max, and a local min at least.
If we give to a particle energy $E=V(x_{min})$ or $E=V(x_{max})$, what is the time it takes to move to $x_{min}/x_{max}$?

I know you can start from the integral of the energy.

$$t-t_{0}=int frac{dx}{sqrt{frac{2}{m}(E-V(x))}}$$

Does anybody know where can I find it?