Verify that \(\mathbf{F}=abla f\) and evaluate the line integral of \(\mathbf{F}\) over the given path.

\(\mathbf{F}(x, y)=\langle 3,6 yangle, \quad f(x, y)=3 x+3 y^{2} ; \quad \mathbf{r}(t)=\left\langle t, 2 t^{-1}ightangle\) on the interval \(1 \leq t \leq 4\)