From: Albert_Rich@msn.com   
      
   On Thursday, November 14, 2013 10:11:17 AM UTC-10, clicl...@freenet.de wrote:   
      
   >> 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   
      
   Ok, for integrands of the form (c+d x)^n/(a+b x) when n is symbolic, the next   
   version of Rubi will return   
      
       (c+d*x)^n/(b*n*(b*(c+d*x)/(d*(a+b*x)))^n)*   
         2F1(-n,-n,1-n,-(b*c-a*d)/(d*(a+b*x)))   
      
   The simpler equivalent rule derived using Euler's transformation is not used   
   since it is harder to simplify its derivative back to the original integrand.   
      
   Note that in addition to examples 6a.n and 6b.n in Timofeev Chapter 8, this   
   change favorably(?) affects example 14. A revised pdf file incorporating these   
   changes for the examples in Chapter 8 is now available at   
      
   http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf   
      
   Albert   
      
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