Explicit inversion of η(γ) from the free-charge relation (a4.15)

Determine an explicit closed-form expression for the unique solution η(γ) to the equation (1 + (3γ η^2)/√(1 − η^3))^2 η = a^4/R^4, which arises in the electrostatic point-charge solution of the nonlinear electrodynamics with Lagrangian density f(s) = s + (γ/β)(1 − √(1 − (2β s)^3)), where a = β^{1/4} q^{1/2} is the effective radius and R > 0 is the radius of the ball around the point charge.

Background

In the electrostatic analysis of the nonlinear model with Lagrangian density f(s) = s + (γ/β)(1 − √(1 − (2β s)^3)), the authors paper the free charge contained in a ball of radius R around an electric point charge. Introducing the variable η = β E_r^2 and the effective radius a = β^{1/4} q^{1/2}, they derive the implicit relation (1 + (3γ η^2)/√(1 − η^3))^2 η = a^4/R^4 for η as a function of the coupling parameter γ.

To proceed analytically, the authors invert this relation only implicitly by expressing γ in terms of η, but explicitly note they cannot solve for η(γ). An explicit closed-form η(γ) would enable exact formulas for the free charge Q_free(R) and energy E(R) distributions, rather than relying on implicit parameterizations or numerical inversion.