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I want to integrate below form

x5^b*(1 - x5)^c*(e + d*x5^0.5 + f*x5)

1./Integrate[x5^b*(1 - x5)^c*(e + d*x5^0.5 + f*x5), {x5, 0, 1}, 
  Assumptions -> {{b, c, e, d, f} \[Element] Reals, c > -1, b > -1}]

and the Mathematica answer is

1./(Gamma[
  1 + c] ((e Gamma[1 + b])/Gamma[2 + b + c] + (1. d Gamma[1.5 + b])/
   Gamma[2.5 + b + c] + (f Gamma[2 + b])/Gamma[3 + b + c]))

but I want see this answer in form of Beta function what should I do?

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1 Answer 1

foo[arg_] := arg /. {Gamma[p_] Gamma[q_] :> Gamma[p + q] Beta[p, q]}

FullSimplify[1./(Gamma[1 + c] ((e Gamma[1 + b])/Gamma[2 + b + c] + (1. d Gamma[1.5 + b])/
   Gamma[2.5 + b + c] + (f Gamma[2 + b])/Gamma[3 + b + c])), 
   TransformationFunctions -> {Automatic, foo}]
 1./(e Beta[1 + b, 1 + c] + 1. d Beta[1.5 + b, 1 + c] + f Beta[2 + b, 1 + c])
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+1, nice work... –  rasher Feb 7 at 10:58
    
@rasher Thanks, but it was wikipedia, because I could not remember the relationship :D –  Sektor Feb 7 at 10:59
    
dear Sektor, thank you so much for your help. you kinda save my life. again thank you –  asal Feb 7 at 11:15
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