I wrote:   
   > Let me recall theory: integrals as above can be reduced to sum   
   > of three different kinds:   
   >   
   > (c+d*x^3)^(1/2)   
   > x*(c+d*x^3)^(1/2)   
   > (x - a)^(-1)*(c+d*x^3)^(1/2)   
      
   Oops, of course in all three above there should be (c+d*x^3)^(-1/2)   
      
   > Note that rules present in Rubi are doing this, so there is no   
   > point at looking triples different than (m, n, p) = (-1, -1, -1).   
   >   
   > First forms above (first and second kind) are no problem: single pattern   
   > for each.  Problem is with third kind.  First, to have any chance   
   > of elementary integral one have to subract apropriate multiple   
   > of integral of first kind.  Second, corrected single term is   
   > elementary if (a, (c + d*a^3)^(1/2)) is a point of finite order   
   > on elliptic curve.  Third, modulo derivatives of elementary functions   
   > multiple terms of form   
   >   
   > (c + d*a_i)^(1/2)*(x - a_i)^(-1)*(c+d*x^3)^(1/2)   
      
   again should contain (c+d*x^3)^(-1/2)   
      
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