Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for the vector field $\mathbf{F} = \langle y e^{xy} + z, x e^{xy}, x \rangle$ along the spiral path parameterized by $\mathbf{r}(t) = \langle t \cos(\pi t), t \sin(\pi t), t^2 \rangle$ for $t \in [0, 1]$.