Eliminate radicals from a symbolic expression

I get a lot of expressions of the type

-(((11 + z) (13 - z + Sqrt[25 - 26 z + z^2]))/(
6 (-11 - z + Sqrt[25 - 26 z + z^2]) (7 - z + Sqrt[25 - 26 z + z^2])))


Which contain radicals in z but are in fact rational functions of z (a-priori). In this case

(11 + z)/(24 + 12 z)


Sometimes Simplify or FullSimplify can find the rational expression, but in more complicated cases they can not. Is there an elegant way to eliminate radicals from an arbitrary expression of this type? I could try to cook up an ansatz for the rational function based on power counting in z or something like that, but maybe there is a ready solution?

• Do you have an example where FullSimplify fails to eliminate radicals? – flinty Aug 7 '20 at 13:13
• It is a good case for Cancel (reference.wolfram.com/language/ref/Cancel.html) – yarchik Aug 7 '20 at 19:37
• @yarchik Cancel does not seem to work even on the example from OP – Weather Report Aug 7 '20 at 19:53
• @flinty I failed to produce one. I thought I have it but maybe it was just Simplify which didn't work. Anyway I deal with very large expressions so the full power of FullSimplify may be too much here. Also it seems I've found a workaround for my particular type of expressions so the question is no longer pressing for me although it could be nice to know the answer. – Weather Report Aug 7 '20 at 19:55