Bounded improper integral

Question

If $x(t)$ is a solution of a globally asymptotically stable second-order ODE, $$\ddot{x}(t) + \alpha \dot{x}(t) + g(x(t)) = 0,$$ where $g(x)$ is an smooth non-linear function which is everywhere continuous and differentiable as well as integrable and $\alpha \in \mathbb{R}_ {>0} $ and if $\|x(t)\|$ is $\mathrm{C}^3$ and $\int_{0} ^{\infty}\tau||\dot{x}(\tau)||^4d\tau < \infty$, then $t||\dot{x}(t)||^2$ is convergent. Are there any exceptions to this inference?