This is an answer to Peter's question in the comments, namely, how did I deduce the radius of convergence to be $\sqrt[3]{27/256}$? I have no proof as the technique is experimental, relying completely on Mathematica. But, this is how I did it.
First, we invert the series using Mathematica's InverseSeries
command.
invSeries = InverseSeries[Series[x + x^4, {x, 0, 30}]]
Next, we use FindSequenceFunction
to generate a closed form expression for the non-zero coefficients.
a[n_] = FullSimplify[FindSequenceFunction[
DeleteCases[CoefficientList[Normal[invSeries], x], 0], n]]
Clearly, this step has some issues. It doesn't always work and, even when it appears to work, there's no guarantee that it works for all $n$. Of course, we can check that the formula works for fairly large $n$, but there's still no proof here. Furthermore, this expression is clearly not as nice as the one found by Raymond!
Now that we have a candidate for the $a_n$s, though, we can easily use a ratio test to find the radius of convergence.
Limit[a[n]/a[n + 1], n -> Infinity]
(* Out: -27/256 *)
Taking the absolute value and accounting for the fact that only every 3rd term is non-zero, we get $\sqrt[3]{27/256}$.