A scalar potential $\varphi$ has the following gradient: $\bigtriangledown \varphi =yz\hat{i}+xz\hat{j}+xy\hat{k}$ . Consider the integral $\int_{c}^{ }\bigtriangledown \varphi .d\overrightarrow{r}$ on the curve $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$.
The curve $C$ is parameterized as follows:$\begin{cases} x=t\\ y=t^2 \text{ and }1\leq t\leq 3 \\ z=3t^2 \end{cases}$.
The value of the integral is ________