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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)

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