Smoothness of integral of a smooth function

Question

If I have a function $l(w,x)$, defined as $\mathbb R^n \times \mathbb R^m \mapsto \mathbb R$ that is continuously differentiable and $L$-smooth with respect to $w$, $\|\nabla l(w,x) - \nabla l(w',x)\| \leq L$ for all $w,w' \in \mathbb R^n$, is the function $w \mapsto \int_{x \in \mathbb R^m} l(w,x) d\mathbb P(x)$ always smooth for a probability measure $\mathbb P$? If not, what are some sufficient conditions to guarantee its smoothness?