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In this case, you can try with

sage: K.<x> = FunctionField(SR)

or

K = FractionField(PolynomialRing(SR, 'x'))
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No.2 Revision

In this case, you can try with

sage: K.<x> = FunctionField(SR)

or

K = FractionField(PolynomialRing(SR, 'x'))

I am not sure how big are your rational functions, but if i do:

sage: f = sum(K.random_element() for i in range(100))
sage: f.numerator() 
...
sage: f.denominator()
....

and this is almost instantaneous.