|\^/|     Maple 7 (SUN SPARC SOLARIS)
._|\|   |/|_. Copyright (c) 2001 by Waterloo Maple Inc.
 \  MAPLE  /  All rights reserved. Maple is a registered trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.

> _Envdiffopdomain;
                                    [DF, x]

> y1,y2 := 1+a*x, x*x+b*x;
                                              2
                          y1, y2 := 1 + a x, x  + b x

> LCLM(DF-diff(y1,x)/y1, DF-diff(y2,x)/y2);
bytes used=1005232, alloc=851812, time=0.36
                     2      (1 + a x) DF            a
                   DF  - 2 -------------- + 2 --------------
                              2                  2
                           a x  + 2 x + b     a x  + 2 x + b

> L1:=%;
                        2      (1 + a x) DF            a
                L1 := DF  - 2 -------------- + 2 --------------
                                 2                  2
                              a x  + 2 x + b     a x  + 2 x + b

> symmetric_product(L1, DF^2);
                                        4
                                      DF


> lprint(L1);
DF^2-2*(1+a*x)/(a*x^2+2*x+b)*DF+2*a/(a*x^2+2*x+b)

homogeneous:

> L1:=DF^2-2*(c+a*x)/(a*x^2+c*2*x+b)*DF+2*a/(a*x^2+c*2*x+b);
                      2       (c + a x) DF              a
              L1 := DF  - 2 ---------------- + 2 ----------------
                               2                    2
                            a x  + 2 c x + b     a x  + 2 c x + b

> symmetric_product(L1, DF^2);
bytes used=2005948, alloc=1507052, time=0.55
                                        4
                                      DF

> L2:=symmetric_product(DF*DF, DF+coeff(L1,DF,1)/4);
                                            2                2  2
              2     (c + a x) DF         5 c  + 6 a x c + 3 a  x  - 2 a b
      L2 := DF  - ---------------- + 1/4 --------------------------------
                     2                             2             2
                  a x  + 2 c x + b             (a x  + 2 c x + b)

> L1:=symmetric_product(L1, DF-coeff(L1,DF,1)/4);
                                           2  2                         2
             2     (c + a x) DF         3 a  x  + 6 a x c + 10 a b - 7 c
     L1 := DF  - ---------------- + 1/4 ---------------------------------
                    2                              2             2
                 a x  + 2 c x + b              (a x  + 2 c x + b)

> symmetric_product(L1,L2);
                                        4
                                      DF

> L1-L2;
        2  2                         2          2                2  2
     3 a  x  + 6 a x c + 10 a b - 7 c        5 c  + 6 a x c + 3 a  x  - 2 a b
 1/4 --------------------------------- - 1/4 --------------------------------
                2             2                        2             2
            (a x  + 2 c x + b)                     (a x  + 2 c x + b)

> %/2;
        2  2                         2          2                2  2
     3 a  x  + 6 a x c + 10 a b - 7 c        5 c  + 6 a x c + 3 a  x  - 2 a b
 1/8 --------------------------------- - 1/8 --------------------------------
                2             2                        2             2
            (a x  + 2 c x + b)                     (a x  + 2 c x + b)

> %^2;
/       2  2                         2          2                2  2        \
|    3 a  x  + 6 a x c + 10 a b - 7 c        5 c  + 6 a x c + 3 a  x  - 2 a b|
|1/8 --------------------------------- - 1/8 --------------------------------|^
|               2             2                        2             2       |
\           (a x  + 2 c x + b)                     (a x  + 2 c x + b)        /

    2

> factor(%);
                                            2 2
                                    (a b - c )
                            9/4 -------------------
                                    2             4
                                (a x  + 2 c x + b)
                                                          
This is the function c(x) in the paper. You see that to make it into a rational
function that depends linearly on (a:b:c) up to a some factor, you need to take
c(x)^(-1/4), which explains how to find the result given in the paper.