Evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$, where $f(x, y, z)=7 x y^{2} z+2 x^{2}$ and $C$ is the curve $x=t^{2}, y=e^{t^{2}-1}, z=t^{2}+t,-1 \leq t \leq 1$

Step 1 ：Define the scalar field \(f = 7x y^{2} z + 2 x^{2}\).

Step 2 ：Compute the gradient of \(f\) to get the vector field \(\mathbf{F} = [4x + 7y^{2}z, 14x y z, 7x y^{2}]\).

Step 3 ：Define the parameterization of the curve \(C\) as \(r = [t^{2}, e^{t^{2}-1}, t^{2} + t]\).

Step 4 ：Compute the differential displacement vector \(d \mathbf{r} = [2t, 2t e^{t^{2} - 1}, 2t + 1]\).

Step 5 ：Compute the dot product of \(\mathbf{F}\) and \(d \mathbf{r}\) to get \(\mathbf{F} \cdot d \mathbf{r} = 28t^{3}(t^{2} + t)e^{2t^{2} - 2} + 7t^{2}(2t + 1)e^{2t^{2} - 2} + 2t(4t^{2} + 7(t^{2} + t)e^{2t^{2} - 2})\).

Step 6 ：Integrate \(\mathbf{F} \cdot d \mathbf{r}\) over \(t\) from -1 to 1 to get the final result.

Step 7 ：The value of the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) is \(\boxed{14}\).