A simple example is given by $$ f(t,x)=\cases{\exp(-(x-t^{-2})^2),&$t\neq 0$,\\0,&$t=0.$} $$ For each fixed $t\neq 0$, $\int f(t,x)\,dx$ is a Gaussian integral equal to $\sqrt{\pi}$, while for $t=0$, the integral equals to zero. Therefore, $F(t)$ is not even continuous at $0$, let alone differentiable.
On the other hand, at $t\neq 0$, any partial derivative of $f$ has a form $P(t^{-1},x)\exp(-(x-t^{-2})^2)$ for some polynomial $P$. This tends to zero as $t\to 0$, uniformly in $x$ in compacts. Therefore, $f$ is smooth with all partial derivatives vanishing on at $\mathbb{R}$. Also, any expression of this form, and in particular $\partial_t f(t,x)$, is clearly integrable over $x$ for all $t$.