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.   
      
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