When does the equation $(A^2+x^2)^sx = C$ have an explicit, closed-form solution? Solving a certain inverse problem reduces to solving the following elementary equation:
$$
(A^2+x^2)^sx = C
$$
Here

*

*$x$ is the unknown. We are interested in $x > 0$.

*$s \in \mathbb{R}$ is a real number.

*$A > 0$ is a fixed real number.

*$C > 0$ is a fixed number, small enough that the equation has at least one solution.

It turns out that depending on the value of $s$ the equation has one or at most two solutions. The left side is strictly increasing when $s \ge -1/2$. It is strictly increasing up to a given point, after which it is strictly decreasing, when $s < -1/2$.
For what values of $s$ does the equation have a closed form solution?
I know of the following cases:

*

*$s = -1$ second order polynomial

*$s = -1/2$ second order polynomial

*$s = 0$ trivial

*$s = 1/2$ fourth order polynomial with special form

*$s = 1$ third order polynomial, which I did not bother to solve

Are there more, and is it possible to verify that all have been found?
Note that the equation is polynomial if $s$ is rational, but not all polynomial equations have explicit solutions.

Because, in the general case, the equation is a polynomial equation in dependence of algebraically independent monomials ($x,(A^2+x^2)^s)$, the equation cannot be solved for $x$ by rearranging it by applying only finite numbers of elementary functions/operations we can read from the equation.
Other tricks, Special functions, numerical or series solutions could help.
$\ $
$$(A^2+x^2)^sx=C$$
For rational $s$, your equation is an algebraic irrational equation or an algebraic equation over the reals and we can use the known solution formulas and methods for such kinds of equations, e.g. for radicals (radical expressions) as solutions.
see e.g. closed-form expression for roots of a polynomial and related posts
for $(A,C,s,x\in\mathbb{R})\land(A,C,x>0)\land(s\neq 0)$:
$$A^2x^\frac{1}{s}+x^{\frac{1}{s}+2}=C^\frac{1}{s}$$
For rational $s\neq 0$, this equation is related to a trinomial equation and we can use the methods for such kinds of equations.
For real $s\neq 0$, the equation is in a form similar to a trinomial equation. A closed-form solution can be obtained using confluent Fox-Wright Function $\ _1\Psi_1$ therefore.
Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104
Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106
