is_constant incorrectly reports False

[smichr]

From #19576,

>>> expr = (9*x + sqrt(3))/(3*sqrt(3)*x + 1)
>>> expr.is_constant
False

but

s = solve(minpoly(expr), x)
v = set([expr.subs(x, i) for i in s])
if len(v) == 1:
    print (v.pop().is_constant)
else:
    print (False)

gives True. Using the extension=True option to cancel also reduces this to a constant:

>>> cancel(expr,extension=True)
sqrt(3)

I'm not sure if there would be cases where the expression could be constant but the extension=True would fail.

I suspect that the false conclusion comes after evaluating expr._random() and comparing this float to the value obtained when using an integer for x.