I'm trying to find a solution in terms of hypergeometric functions or a closed-form solution to the equation: $$ x^5 + a x^3 - b x^2 + c = 0 $$ where $a,b,c>0$. I've tried converting the equation to Bring-Jerrard form with the following Tschirnhaus transformations: $$ x = y + h $$ $$ x = y + \alpha y^2 + \beta y^{3} $$ to $$ x = \alpha + \beta y + \gamma y^{2} + \delta y^{3} $$ but none of them have been able to cancel the quadratic term (or other complications arise). Any suggestions or recommendations? I'm not looking for approximate (or numerical) solutions.
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1You mean $\delta y^3$ and not $\theta y3$. See the references here, how to reduce to $x^5+ax+b=0$. – Dietrich Burde May 31 '25 at 18:20
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Another 4 term quintic’s inverse is solved here – Тyma Gaidash Jun 01 '25 at 02:59