read(diffop);

g:=DF^3-4*(2*x-1)/(x-1)/x*DF^2+1/25*(749*x^2-749*x+249)*DF/(x^2-2*x+1)/x^2-
9/50*(266*x^3-399*x^2+265*x-66)/x^3/(x^3-3*x^2+3*x-1);

sy2:=symmetric_power(g,2);
# g is irreducible, so the second symmetric power is completely reducible,
# i.e. sy2 is the LCLM of irreducible operators. We can compute an LCLM
# factorization by computing the eigenring:

v:=eigenring(sy2);

# Now v contains a non-constant r:
r:=v[1] ; if r=1 then r:=v[2] fi;

# We can compute the eigenvalues of this endomorphism r as follows:

ev:={solve(endomorphism_charpoly(sy2,r))};

# r has 2 different eigenvalues. Using these eigenvalues we can compute the
# following right hand factors of sy2:

R1:=GCRD(sy2,r-ev[1]);
R2:=GCRD(sy2,r-ev[2]);

# Since the dimension of the eigenring is 2, there can not be more right
# hand factors of sy2. So R1,R2 is are irreducible and sy2=LCLM(R1,R2);
# Lets check this latter statement:

normal(sy2-LCLM(R1,R2));
# =0, OK.

# If you think this took a long time, then try to compute the eigenring of
# the third symmetric power of g (takes about 250 times longer...)
# It costs 30 kilobytes to express (with the command lprint) the output of
# eigenring(symmetric_power(g,3)). Its dimension is 3, which (using the
# fact that g is irreducible) means that symmetric_power(g,3) factors into
# 3 irreducible factors.