Solution to $(A+x^2)e^x=B$ with Lambert W function Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?
 A: You seek a solution in $x$ of the transcendental equation
$$e^x(x-t_1)(x-t_2)=a.$$
(The coefficients $t_1,t_2$ are real for $A<0$, complex otherwise.) The solution $W(t_1,t_2;a)$ is referred to as the "quadratic Lambert-W function". It is
studied in several recent papers:

*

*General Relativity and
Quantum Mechanics: Towards a Generalization of the Lambert W
Function (2006)

*On the
generalization of the Lambert W function with applications in
theoretical physics (2014) [section 4]

*Some physical applications of generalized Lambert functions (2015)

*Generalization of Lambert W-function, Bessel polynomials and transcendental equations (2015)

A series expansion is presented in
Asymptotic series of Generalized Lambert W Function (see also
this MO posting).
A: The solution to this transcendental algebraic equation is indeed a Generalized Lambert W function and there are a number of papers on the latter, not only the pioneering 2006 AAECC paper mentioned here.  See the papers in https://www.researchgate.net/project/Generalized-Lambert-W-function
Granted the second reference, the one by Mezo and Keady claims that the approach used for a particular instance in Physics is 'unsatisfactory' for getting all the solutions but that's an issue of completeness of solutions - not an actual error. Subsequent papers in SIGSAM in that link I just gave, offer more solutions.  E.g. check out the most recent publications by Aude Maignan on the ResearchGate project:
https://www.researchgate.net/project/Generalized-Lambert-W-function
Much work has been done since the 2006 paper.
