Albert Rich schrieb:   
   >   
   > On Wednesday, November 13, 2013 1:56:55 AM UTC-10, clicl...@freenet.de wrote:   
   >   
   > > The hypergeometric series here (and in Example 3) terminate for a   
   > > non-negative integer exponent n, while the series for your evaluation of   
   > > Examples 6a,b terminate (after applying Euler's transformation) for a   
   > > negative integer exponent n. The latter type of representation is   
   > > somewhat more compact; a disadvantage is that the antiderivative 6a for   
   > > positive a,m,x ends up on the branch cut of the hypergeometric function.   
   > > I therefore propose to normalize to the former type:   
   >   
   > For the integral of (1+x)^n/x, Rubi currently returns   
   >   
   > -(1+x)^(1+n) * 2F1(1,1+n,2+n,1+x) / (1+n)   
   >   
   > In light of your comments above, would it be better to return   
   >   
   > (1+x)^n * 2F1(-n,-n,1-n,-1/x) / (n*(1+1/x)^n)  ?   
   >   
      
   I think so. For positive x and non-integer n you are no longer sitting   
   right on the edge of a cliff then - the thought alone makes me dizzy. An   
   equivalent (by Euler's transformation) but simpler antiderivative is:   
      
     (1+x)^(1+n) * 2F1(1,1,1-n,-1/x) / (n*x)   
      
   Note that the singularity at x=0 is already present in the integrand.   
   And Pfaff's transformation of these two puts one on the brink of the   
   chasm when x is negative and small:   
      
     (1+x)^n * 2F1(-n,1,1-n,1/(1+x)) / n   
      
   Martin.   
      
   PS: I have seen your "piecewise constants" called "differential   
   constants" by WRI's Daniel Lichtblau, but the former term is much much   
   more popular according to Google.   
      
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