From: Albert_Rich@msn.com   
      
   On Sunday, January 28, 2018 at 7:37:33 PM UTC-10, clicl...@freenet.de wrote:   
   > Albert Rich schrieb:   
   > >   
   > > I would like to be able to assure users that Rubi will find at least   
   > > near optimal antiderivatives for all integrands of the form   
   > >   
   > >     x^m*(a+b*x^3)^p*(c+d*x^3)^q   
   > >   
   > > where m and p are integers, and q is a half-integer. For that I need   
   > > to know:   
   > >   
   > > In light of the ongoing theoretical discussion on this thread, is the   
   > > antiderivative of x/((b+x^3)*sqrt(1+x^3)) elementary for any values of   
   > > b other than 4 and -8. If so, can someone give me an example and its   
   > > elementary antiderivative.   
   > >   
   >   
   > Testing for standard Goursat cases (Eq. 5 on p. 111 of Goursat's 1887   
   > paper) leads to (b + 8)*(b^2 - 20*b - 8) = 0, any such b suffices for   
   > an elementary antiderivative, which FriCAS should be able to determine.   
   > I have not implemented the material in paragraphs 4 and 5 of Goursat's   
   > paper, which is special to the radical SQRT(x^3-1), but nothing new   
   > seems to follow from it (Goursat may not have been aware of the fact   
   > that his eq. 5 alone suffices to detect b = -8).   
   >   
   > Testing for Goursat cases of the equivalent cube-root integrand leads   
   > to (b + 8)*(b - 4) = 0, as mentioned in a post of Sun, 21 August 2016   
   > 18:12:49 +0200 already (the thread was entitled "Rubi 4.9.2 do not   
   > integrate Timofeev #319 while 4.9 does"). Such tests of cube-root   
   > integrands do not appear in Goursat's paper (or elsewhere to my   
   > knowledge), but the case of b = 4 apparently goes undetected otherwise.   
   >   
   > Needless to say, the cases of b = 10 +- 6*SQRT(3) result in new cube-   
   > root pseudo-elliptics as well.   
   >   
   > For any possible further (i.e. non-Goursat) pseudo-elliptic instances   
   > of your integrand, somebody more familiar with elliptic curves must be   
   > consulted.   
   >   
   > Martin.   
      
   Knowing that x/((10+6*sqrt(3)+x^3)*sqrt(1+x^3)) has an elementary   
   antiderivative, how does one goes about finding it? I was not able to get   
   FriCAS to help out...   
      
   Albert   
      
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