The Wolfram Language can efficiently handle both univariate and multivariate rational functions, with built-in functions immediately implementing standard algebraic transformations.

Apart — decompose into partial fractions with minimal denominators

Together — put rational functions over a common denominator

Cancel — cancel common factors between numerator and denominator

Numerator, Denominator — pick out numerator, denominator

NumeratorDenominator  ▪  PolynomialRemainder  ▪  PolynomialQuotient

Factor — factor numerator and denominator, canceling common factors

Expand — expand everything except denominators

ExpandAll — expand everything, including denominators

ExpandNumerator, ExpandDenominator — expand numerators and denominators only

PadeApproximant — find a rational approximation to any function

Operations on Rational Functions

Residue  ▪  Integrate  ▪  Sum