When given a tenth degree polynomial $$f(x)=\sum_{n=0}^{10} a_n x^n$$ I wish to compute an inverse via the Lagrange Inversion Theorem to produce a generalized hypergeometric sum similar to a Bring Ultradical or Kampe de Feriet Function.

I believe that I have been able to compute an inverse for a function of the form $$\hat{f}(y)= y^{10} + \sum_{n \in S} b_n y^n + b_0$$ Where $S\subset\{1,...,9\}$ contains no more than three elements. For example, $S = \{1, 5, 6\}$.

Tschirnhaus asserted that two of the intermediate terms for an $n^{\text{th}}$ degree polynomial could be removed by means of his transformation in his original paper. I am also aware, that a higher order substitution must be made to convert a polynomial into Bring-Jerrard form.

Question: Is there a substitution via a rational function $y(x)$ that could convert $f(x)$ to $\hat{f}(y)$?

I was hoping to express $y(x)$ as a $5^{\text{th}}$ degree polynomial, which would allow me to recover $x$ using Bring ultradicals. I had hoped then, that I could nest a Bring radical (in terms of one variable) inside of a hypergeometric series (in terms of four variables) as opposed to examining one higher order hypergeometric series.