Suppose you have the following integral
$$V_{pq}(t):=\int_{\mathbb{R}^3}\frac{\chi_p(x,t)\chi_q(x,t)}{\|x-R_c(t)\|}\mathrm{d}x,$$
where $\chi_p(x,t):= (x_1-R_{p,x_1}(t))^\ell(x_2-R_{p,x_2}(t))^m(x_3-R_{p,x_3}(t))^ue^{-\|x-R_p(t)\|²}$ and $\chi_q(x,t):= (x_1-R_{q,x_1}(t))^n(x_2-R_{q,x_2}(t))^v(x_3-R_{q,x_3}(t))^se^{-\|x-R_q(t)\|²}$, with $\ell, m, u, n, v, s \in \mathbb{N}$, $(x_1,x_2,x_3)=x, (R_{j,x_1}(t),R_{j,x_2}(t),R_{j,x_3}(t))=R_j(t)$ and $t\mapsto R_c(t), t\mapsto R_q(t), t\mapsto R_p(t)$ are all smooth curves from an shared bounded interval $I$ into $\mathbb{R}^3$.
Is $V_{pq}:I \longrightarrow \mathbb{R}$ under certain conditions on the curves a smooth function?
