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There was a lengthy series of posts [link], in late 2008, concerning the paper of Demetrios Panyotounakos that claimed to have solved a large class of Abel equations.  As I read the posts, it would seem that the final conclusion had not yet occurred, as to the validity of his claim. 

In particular there were questions to Edgardo Cheb-Terrab as to why this...

Usually Maple gives solutions in terms of radicals only up to degree=4
(for example using RootOf + allvalues).

Using 'irreduc' (to test first) and 'galois' (to check for the Galois
group, if degree <= 9) I have cases, where the result implies, that by
theory the roots can be given through radicals:

Though the results are difficult to read in my case they are 'C(6)' or
'C(8)', the cyclic groups of that order - thus abelian and IIRC those
groups are solvable.