From: hrubin@skew.stat.purdue.edu   
      
   On 2014-06-30, paulandrewbird@gmail.com  wrote:   
   > Hi, I'm trying to solve the integral:   
      
   > integral( exp( a*x^4+4*b*x^3*y+6*c*x^2*y^2+4*d*x*y^3+e*y^4)   
   ,x=-infty..infty, y=-infty..finty)   
      
   > I tried on Wolfram alpha but it was too complicated for it.   
      
   > I'm guessing it's going to be in the form 1/P(a,b,c,d,e)^(1/8) where   
   P is a 4th degree polynomial in the coefficients of the quartic.   
      
   > My guess is that P is the discriminant of the polynomial but that is   
   just a guess.   
      
   > Is there any good free software that can solve this? Can Mathematica   
   or Maple solve this? Can you?   
      
   The easiest way that I can see is to find the area A where the   
   homogeneous quartic is >= -1.  Then the integral is A times the   
   integral of exp(-x^4), which is Gamma(1+1/4).  The same approach   
   is the best I can think of for any homogeneous expression.   
      
   > Thanks.   
      
      
   --   
   This address is for information only.  I do not claim that these views   
   are those of the Statistics Department or of Purdue University.   
   Herman Rubin, Department of Statistics, Purdue University   
   hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)