From: nma@12000.org   
      
   On 4/9/2020 6:07 PM, antispam@math.uni.wroc.pl wrote:   
      
   >> integrate(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+7*x   
   10+8*x^12)^(1/2))/(x^9*(-2+x^6)),x);   
   >> ?   
   >> ?? >> Error detected within library code:   
   >> ?? integrate: implementation incomplete (residue poly has multiple   
   non-linear factors)   
   >   
      
   > The two above are known problems, each will take more work to fix.   
   > Actually, the first one requires reorganization of integration   
   > routines.  The second one is king of open problem: known methods   
   > of solving may require prohibitivly long time and nobody knows   
   > of really efficient method.   
   >   
      
   The above comes down to separate 4 integrands   
      
   (8*Sqrt[32 - 14*x^4 - 32*x^6 - 4*x^8 + 7*x^10 + 8*x^12])/x^9   
      
   (2*Sqrt[32 - 14*x^4 - 32*x^6 - 4*x^8 + 7*x^10 + 8*x^12])/x^5   
      
   (2*Sqrt[32 - 14*x^4 - 32*x^6 - 4*x^8 + 7*x^10 + 8*x^12])/x^3 +   
      
   (3*x*Sqrt[32 - 14*x^4 - 32*x^6 - 4*x^8 + 7*x^10 + 8*x^12])/(-2 + x^6)   
      
   It seems to boild down to integrand of this pattern   
      
              sqrt(a + b*x^4 + c*x^6 + d*x^8 + e*x^10 + f*x^12)   
      
   ======================   
   8) -> integrate( sqrt(a + b*x^4 + c*x^6 + d*x^8 + e*x^10 + f*x^12),x)   
      
               x  +--------------------------------------+   
             ++   |  12      10      8      6      4   
       (8)   |   \|%A  f + %A  e + %A d + %A c + %A b + a d%A   
            ++   
   ========================   
      
   Then I tried a simpler one with polynomial of just degree 4 under the sqrt   
      
          integrate( sqrt(a + b*x + c*x^2 + d*x^3 + e*x^4 ),x)   
      
   But it could not do it either. But Mathemtica managed to  do this one   
   but it uses special functions EllipticF, EllipticPi, and result too   
   large to post. These look like really hard integrals.   
      
   Is the full Risch algorithm supposed to be able to handle these also?   
      
   --Nasser   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)