Can this fraction be simplified ?

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For another perspective : symbolic expression are too wide so that equaity could not be decided, in particular there can not be a consistent simplification procedure.

However, the expression you are dealing with represents an algebraic number. The field of algebraic numbers is a safer place : the problems above become decidable. So, let me suggest the following approach;

sage: a = (sqrt(2)+2)/(sqrt(2)+1)
sage: a.parent()
Symbolic Ring
sage: b = QQbar(a)
sage: b
1.414213562373095?
sage: b.parent()
Algebraic Field
sage: b.radical_expression()
sqrt(2)

Note howewer that the radical_expression method is a bit hackish and does not handle the wohle Galois theory (yet).

As a side remark, both SymPy and Giac are able to simplify it:

sage: s = (sqrt(2)+2)/(sqrt(2)+1)
sage: s._sympy_().simplify()
sqrt(2)
sage: s._giac_().simplify()
sqrt(2)

EDIT: regarding the large fraction involving the symbolic variable x, only Giac is able to simplify it:

sage: f = 3*(x^4+4*sqrt(3)*(x^2+6)*sqrt(x^2+3)+24*x^2+72)/(sqrt(3)*(x^5+24*x^3+72*x)+12*(x^3+6*x)*sqrt(x^2+3))
sage: f._sympy_().simplify()
(3*x**4 + 72*x**2 + 3*sqrt(3*x**2 + 9)*(4*x**2 + 24) + 216)/(x*(12*sqrt(x**2 + 3)*(x**2 + 6) + sqrt(3)*(x**4 + 24*x**2 + 72)))
sage: f._giac_().simplify()
sqrt(3)/x

Well... The default Sage algorithms aren't always the most efficient.However, one can try :

sage: import sympy
sage: sympy.sympify((sqrt(2)+2)/(sqrt(2)+1)).simplify()._sage_()
sqrt(2)

We also have :

sage: ((sqrt(2)+2)/(sqrt(2)+1)/sqrt(2)).canonicalize_radical()
1

as well as :

sage: from giacpy_sage import *
sage: libgiac.simplify((sqrt(2)+2)/(sqrt(2)+1)).sage()
sqrt(2)

(this one can probably be done more economically...).

HTH...