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What happens with your big rational function if you work in the FunctionField instead of symbolic functions ? For example:

sage: K.<x> = FunctionField(QQ); K
Rational function field in x over Rational Field
sage: f = K(3*x/(4*(x^2+1))+ x^2/(3*x^3-5))
sage: f.numerator()
13*x^4 + 4*x^2 - 15*x
sage: f.denominator()
12*x^5 + 12*x^3 - 20*x^2 - 20

What happens with your big rational function if if, instead of working with symbolic functions, you work in the FunctionField instead of symbolic functions ? For example:?

sage: K.<x> = FunctionField(QQ); K
Rational function field in x over Rational Field
sage: f = K(3*x/(4*(x^2+1))+ x^2/(3*x^3-5))
sage: f.numerator()
13*x^4 + 4*x^2 - 15*x
sage: f.denominator()
12*x^5 + 12*x^3 - 20*x^2 - 20

sage: K = FractionField(PolynomialRing(QQ,'x'))
sage: f = K(3*x/(4*(x^2+1))+ x^2/(3*x^3-5))
sage: f.numerator()
13*x^4 + 4*x^2 - 15*x
sage: f.denominator()
12*x^5 + 12*x^3 - 20*x^2 - 20

What happens with your big rational function if, instead of working with symbolic functions, you work in the FunctionField ?

sage: K.<x> = FunctionField(QQ); FunctionField(QQ) ; K
Rational function field in x over Rational Field
sage: f = K(3*x/(4*(x^2+1))+ x^2/(3*x^3-5))
sage: f.numerator()
13*x^4 + 4*x^2 - 15*x
sage: f.denominator()
12*x^5 + 12*x^3 - 20*x^2 - 20

Or in the FractionField ?

sage: K = FractionField(PolynomialRing(QQ,'x'))
FractionField(PolynomialRing(QQ,'x')) ; K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: f = K(3*x/(4*(x^2+1))+ x^2/(3*x^3-5))
sage: f.numerator()
13*x^4 + 4*x^2 - 15*x
sage: f.denominator()
12*x^5 + 12*x^3 - 20*x^2 - 20

Does one of them work ? Which is faster ? Otherwise, could you give us an example of "big rational function" ?