Preamble (updated)
My course notes have
Now consider a closed volume $V$ in the fluid which is bounded by a surface $S$. Let $V$ move with the fluid, so no fluid moves in or out of $V$. [...] Now $\rho\text{ d}V$ is the mass of a fluid element and does not change as the element moves around. Thus $$\frac{d}{dt}\int_V\rho u_i\text{ d}V=\int_V\left(\frac{Du_i}{Dt}\right)\rho\text{ d}V.\tag{$\ast$}$$
Earlier it was stated that
In this course we will be assuming that fluids are incompressible
and the substantive derivative $\frac{D}{Dt}$ was defined, with a note that
Sometimes it is written as $\frac{d}{dt}$.
Question (updated)
I gather from TedShifrin's helpful comments that $V$ is not independent of $t$. So that leaves me wondering how the (or a) derivative operator gets inside the integral. It's not simply undoing the integration, and if it's doing something fancier I can't immediately see what that is. Could anyone give me a pointer on that?
Question (original - ignore if you like)
I see that $V$ doesn't depend on $t$, so we can take the differentiation operator inside the integral to get \begin{align} \frac{d}{dt}\int_V\rho u_i\text{ d}V & = \int_V\frac{d}{dt}(\rho u_i)\text{ d}V \\ & = \int_Vu_i\frac{d\rho}{dt}+\rho\frac{du_i}{\text{d}t}\text{ d}V \\ & = \int_V\frac{du_i}{dt}\rho\text{ d}V, \text{ since } \rho=\text{const.} \end{align}
For this to align with $(\ast)$, I think either $\frac{d}{dt}$ must be the substantive derivative $\frac{D}{Dt}$, for some reason being written in both forms in the same equation, or we must have \begin{align} & \begin{aligned}[t] \frac{du_i}{dt} & = \frac{Du_i}{Dt} \\ & = \frac{\partial u_i}{\partial t}+\textbf{u}\cdot\nabla u_i \end{aligned} \\[1em] \therefore\;& \frac{du_i}{dt} = \frac{\partial u_i}{\partial t}, \hspace{1em} \textbf{u}\cdot\nabla u_i = 0. \end{align}
Further down, $\textbf{u}\cdot\nabla u_i$ is used in a way that I think means it doesn't have to be zero in this context, which seems to point to the use of both notations for the substantive derivative in the same equation. Is that what's happened?