Mfcttrf

documentclassbeamer
newcommandFontvifontsize24selectfont
usepackage[utf8]inputenc
usepackage[compat=1.1.0]tikz-feynman
usepackageamsmath, amsthm, amssymb,amsfonts
usepackagegraphicx
usepackagetikz
usepackagetcolorbox
usepackageamsmath
usepackageamsthm
usepackageamssymb
usepackagetikz-feynman
setcounterMaxMatrixCols20
%usepackagebreqn
usepackagetikz-feynman 
usepackageamsmath
newtheoremrulesRule
usethemeAntibes
usecolortheme
titleNegative Dimensional Integral and Modified Method of Brackets
author[Sumit Banik]textbf Work By-Sumit Banik\ footnotesize Supervised by: B Ananthanarayan
instituteIndian Institute of Science\Bangalore- 560012
date
begin document
begin frame
beginrules labelRule2
beginequation labeleq:rule-B
 beginsplit
 &(A_1 + cdots + A_J)^alpha
 \
 &quad=
 frac1Gamma(-alpha)
 int_c_1-iinfty^c_1+iinftyfracdz_12pi i
 cdots
 int_c_J-iinfty^c_J+iinftyfracdz_J2pi i
 ,
 <z_1 + cdots + z_J - alpha>
 ,
 A_1^z_1
 cdots
 A_J^z_J
 ,
 Gamma(-z_1)
 cdots
 Gamma(-z_J),.
 endsplit 
 endequation
endrules
end frame
end document

I would go for a multline environment. Unrelated: don't load packages twice, and also, if you load amssymb, needless to load amsfonts.

documentclassbeamer
newcommandFontvifontsize24selectfont
usepackage[utf8]inputenc
usepackage[compat=1.1.0]tikz-feynman
usepackagemathtools, amsthm, amssymb
usepackagegraphicx
usepackagetikz
usepackagetcolorbox
usepackagetikz-feynman
setcounterMaxMatrixCols20
usepackagetikz-feynman
newtheoremrulesRule
usethemeAntibes
usecolortheme
titleNegative Dimensional Integral and Modified Method of Brackets
author[Sumit Banik]textbf Work By-Sumit Banik\ footnotesize Supervised by: B Ananthanarayan
instituteIndian Institute of Science\Bangalore- 560012
date

begin document

begin frame
beginrules labelRule2
beginmultlinelabeleq:rule-B
 (A_1 + cdots + A_J)^alpha = 
 \
 frac1Gamma(-alpha)
 int_c_1-iinfty^c_1+iinftyfracdz_12pi i
 cdots
 int_c_J-iinfty^c_J+iinftyfracdz_J2pi i
 ,
 langle z_1 + cdots + z_J - alpharangle times \
 A_1^z_1
 cdots
 A_J^z_J
 ,
 Gamma(-z_1)
 cdots
 Gamma(-z_J)phantomtimes
 endmultline
endrules
end frame

end document 

I think the equation could benefit from a couple of additional line breaks.

documentclassbeamer
usethemeAntibes
usepackageamsmath, amsthm, amssymb%,amsfonts}
newtheoremrulesRule
usepackagegraphicx
usepackage[utf8]inputenc

%% I've commented out all instructions that are repetitive or 
%% do not affect the body of the sample document.
%newcommandFontvifontsize24selectfont
%usepackage[compat=1.1.0]tikz-feynman
%usepackagetikz
%usepackagetcolorbox
%usepackageamsmath
%usepackageamsthm
%usepackageamssymb
%usepackagetikz-feynman
%setcounterMaxMatrixCols20
%usepackagebreqn
%usepackagetikz-feynman 
%usepackageamsmath
%usecolortheme
%titleNegative Dimensional Integral and Modified Method of Brackets
%author[Sumit Banik]textbf Work By-Sumit Banik\ footnotesize Supervised by: B Ananthanarayan
%instituteIndian Institute of Science\Bangalore- 560012
%date

begindocument
beginframe
beginrules labelRule2
beginequation labeleq:rule-B
beginsplit
 &(A_1 + dots + A_J)^alpha \
 &quad= frac1Gamma(-alpha)
 int_c_1-iinfty^c_1+iinftyfracdz_12pi i
 dotsm
 int_c_J-iinfty^c_J+iinftyfracdz_J2pi i
 \
 &quad< z_1 + dots + z_J - alpha\
 &quad> A_1^z_1dotsm A_J^z_J
 ,Gamma(-z_1)dotsmGamma(-z_J),.
endsplit 
endequation
endrules
endframe
enddocument

Addendum, inspired by a detail in @Bernard's parallel answer: If < and > do not denote "less than" and "greater than" and, instead, are supposed to be interpreted as angle brackets (langle and rangle, respectively, in TeX parlance), it would be better to combine rows three and four of the answer shown above into a single line:

documentclassbeamer
usethemeAntibes
usepackageamsmath, amsthm, amssymb
newtheoremrulesRule
usepackagegraphicx
usepackage[utf8]inputenc

begindocument
beginframe
beginrules labelRule2
beginequation labeleq:rule-B
beginsplit
 &(A_1 + dots + A_J)^alpha \
 &quad= frac1Gamma(-alpha)
 int_c_1-iinfty^c_1+iinftyfracdz_12pi i
 dotsm
 int_c_J-iinfty^c_J+iinftyfracdz_J2pi i \
 &qquadtimes 
 langle z_1 + dots + z_J - alpharangle
 ,A_1^z_1dotsm A_J^z_J 
 ,Gamma(-z_1)dotsmGamma(-z_J),.
endsplit 
endequation
endrules
endframe
enddocument

You should split your equation. Please Consider the following code:

documentclassbeamer
usepackage[compat=1.1.0]tikz-feynman
usepackageamsmath, amsthm, amssymb,amsfonts
usepackagetikz
newtheoremrulesRule
usethemeAntibes
begin document
begin frame
beginrules labelRule2
beginalign labeleq:rule-B
 &(A_1 + cdots + A_J)^alpha
 nonumber\
 quad=&
 frac1Gamma(-alpha)
 int_c_1-iinfty^c_1+iinftyfracdz_12pi i
 cdots
 int_c_J-iinfty^c_J+iinftyfracdz_J2pi i
 ,nonumber\
 &<z_1 + cdots + z_J - alpha>
 ,
 A_1^z_1
 cdots
 A_J^z_J
 ,
 Gamma(-z_1)
 cdots
 Gamma(-z_J),. 
 endalign
endrules
end frame
end document