The examples in this subsection have been used for testing purposes by Matthias Goldgruber. He uses symbolic computation on fractions of multivariate polynomials implemented by Stefan Karnel.

Add the terms and simplify them as much as possible (by cancelling etc.) ...

No.1 2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 + 5/54 = ...

This simplification results in large numerals in a small term (can you imagine, how much effort was required for inventing such an example -- just to enjoy the student upon successful simplification !).
No.2 ((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2 = ...

Like all the others on this page, this double fraction results from the fact, that ':' is not defined as the division-operator in Isabelle.  The fraction simplifies considerably, too
No.3 "(x^2 / (1 - x^2) + 1) / (x / (1 - x) + 1) * (1 + x)"  = ...

Another double fraction, which actually simplifies to 1 !
No.4 (36 * b ^ 3 + 3 * a ^ 2 * b ^ 2 + 16 * a ^ 4 * b) / (9 * a ^ 4) = ...

This is much work without such an improblable simplification.
No.5 (15*a^2 / x^3 - 5*b^4 / x^2 + 25*c^2 / x) * (x^3 / (5*a*b^3*c^3)) + 1 / c^3 * (b*x / a - 3*a / b^3) = ...

This is nice, again !
No.6 (a/2 + b/3)*(b/3 - a/2) = ...

And this example probably should be handled differently.
No.7 (a^2/9 + 2*a/(3*b) + 4/b^2)*(a/3 - 2/b) + 8/b^3 = ...

Again, there is a nicer simplification, which does not conform to the normal-form created by ISACs simplifier (which expands nominator and denominator). The topic of 'normal-forms' is of great importance in symbolic computation !
No.8  (x / (5*x + 4*y) - y / (5*x - 4*y) + 1) * (25*x^2 - 16*y^2) = ...

A challenge for beginners !
No.9 (2*x^2 / (3*y) + x / y^2) * (4*x^4 / (9*y^2) + x^2 / y^4) * (2*x^2 / (3*y) - x / y^2) = ...

... and these are just a lot of work, if done by hand ...
No.10 (4*x / (3*y) + 2*y / (3*x))^2 - (2*y / (3*x) - 2*x / y) * (2*y / (3*x) + 2*x / y) = ...
No.11 (15*a^4 / (a*x^3) - 5*a*((b^4 - 5*c^2*x) / x^2)) * (x^3 / (5*a*b^3*c^3)) + a / c^3 * (x*(b/a) - 3*b*(a/b^4)) = ...