It sucks that Danish basically never comments because even after seeing the roots Charlie found with brute force I still don't see what substitution is needed.
(-1 + sqrt(13))/6 is a double root and (-1 - sqrt(61))/30 is a single root.
I also note that you can get an ordinary 6th degree polynomial by simply squaring each side and cleaning up. But that polynomial will have extraneous roots, the other roots are for the negative root of the radical in the original equation.
So then the conjugates (-1 - sqrt(13))/6 (double root) and (-1 + sqrt(61))/30 (single root) should be the other three roots of the 6th degree polynomial.
So maybe the trick is differentiation and then polynomial GCD instead of substitution? Since differentiation will preserve one member a double root.
Thoughts
