horizontal \smash-like command in math mode

You can use the \mathrlap command from mathtools.

\documentclass[a4paper]{report}
\usepackage{mathrsfs}
\usepackage{braket}
\usepackage{mathtools}

\begin{document}

\begin{align}
& \int_{\mathscr{B}, \mathscr{C}} db \, dc \, \ket{a', b, c} \bra{a', b, c} \rightarrow \ket{a'} \bra{a'}, \\
& \int_{\mathrlap{\mathscr{B}, \mathscr{C}}} db \, dc \, \ket{a', b, c} \bra{a', b, c} \rightarrow \ket{a'} \bra{a'},
\end{align}

\end{document}

Four possibilities:

In (1) the width of the subscript is made zero; a \, is needed anyway to separate the differential from the integral sign.

In (2) the subscript is given a nonzero width.

In (3) a manual backspacing is applied.

In (4) the problem is solved in possibly the clearest way: you risk to make your formula unreadable otherwise, because the readers might mistake the subscript with something beneath the first differential.

\documentclass[a4paper]{report}
\usepackage{amsmath} % don't forget
\usepackage{mathtools} % for \mathmakebox
\usepackage{mathrsfs}
\usepackage{braket}

\begin{document}

\begin{equation}
\int_{\mathrlap{\mathscr{B}, \mathscr{C}}}\, db \, dc \, \ket{a', b, c} \bra{a', b, c}
\end{equation}

\begin{equation}
\int_{\mathmakebox[0.5em][l]{\mathscr{B}, \mathscr{C}}} db \, dc \, \ket{a', b, c} \bra{a', b, c}
\end{equation}

\begin{equation}
\int_{\mathscr{B}, \mathscr{C}}\mspace{-18mu} db \, dc \, \ket{a', b, c} \bra{a', b, c}
\end{equation}

\begin{equation}
\int\limits_{\mathscr{B}, \mathscr{C}} db \, dc \, \ket{a', b, c} \bra{a', b, c}
\end{equation}

\end{document}