Counterexample: is there a solution of $\dot{x}(t)=-\nabla f(x)(t)$ with infinite length?

Question

Let $U\subseteq\mathbb{R}^n$ a bounded domain, $f:U\to \mathbb{R}$ is $C^1$ non negative and $x(t)$ a $C^1$ solution of $$ \dot{x}(t) = -\nabla f(x(t)),\quad x(0) = x_0$$ Always exists $C>0$ such that $$ \int \|\dot{x}(t)\|dt \leq C $$ uniformly in $x_0\in U$? If $U$ is not bounded, then it is easy to find counterexamples.