Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.

\(\mathbf{F}(x, y)=\left\langle x^{2} y, y^{2} z, z^{2} xightangle\), the path \(\mathbf{r}(t)=\left\langle e^{-t}, e^{-2 t}, e^{-3 t}ightangle\) for \(0 \leq t<\infty\)