Here are two solutions; one with empheq and a custom box, based on \fcolorbox, the other with the standard \boxed command from amsmath and the multlined environment (requires mathtools – which is loaded by empheq).
\documentclass{article}
\usepackage{geometry}
\usepackage{empheq, amssymb}
\usepackage[svgnames, table]{xcolor}
\DeclareMathOperator{\Res}{\mathbf{Res}}
\newcommand*\widefbox[2][black]{\setlength{\fboxrule}{0.8pt}\setlength{\fboxsep}{1.2ex}\fcolorbox{#1}{white}{\enspace #2\enspace}}
\begin{document}
\begin{empheq}[box={\widefbox[SeaGreen]}]{align*}
I=\frac{1}{|\det \mathbf{A}|} \quad \smashoperator{\sum_{m_{s+1}\cdots m_{r}}} \phi_{m_{s+1}\cdots m_{r}}\Res\biggl[ f(\Bar{n}_{1},\dots,\Bar{n}_{s},n_{s+1}+m_{s+1},\dots,n_{r}+m_{r})\Gamma(-\Bar{n}_{1})\dotsm\Gamma(-\Bar{n}_{s}) & \\
\cdot \frac{\Gamma(1-n_{s+1})\Gamma(1+n_{s+1})\dotsm\Gamma(1-n_{r})\Gamma(1+n_{r})\Gamma(1+m_{s+1})\cdots\Gamma(1+m_{r})}{n_{s+1}\cdots n_{r}\quad\Gamma(1+n_{s+1}+m_{s+1})\cdots\Gamma(1+n_{r}+m_{r})}\biggr] &
\end{empheq}
\[ \boxed{\begin{multlined}
I=\frac{1}{|\det\mathbf{A}|}\quad \smashoperator{\sum_{m_{s+1}\cdots m_{r}}}\phi_{m_{s+1}\cdots m_{r}}\Res\biggl[ f(\Bar{n}_{1},\dots,\Bar{n}_{s},n_{s+1}+m_{s+1},\dots,n_{r}+m_{r})\Gamma(-\Bar{n}_{1})\dotsm\Gamma(-\Bar{n}_{s}) \\
\frac{\Gamma(1-n_{s+1})\Gamma(1+n_{s+1})\dotsm\Gamma(1-n_{r})\Gamma(1+n_{r})\Gamma(1+m_{s+1})\cdots\Gamma(1+m_{r})}{n_{s+1}\cdots n_{r}\quad\Gamma(1+n_{s+1}+m_{s+1})\cdots\Gamma(1+n_{r}+m_{r})}\biggr]
\end{multlined}} \]
\end{document}
