I am trying to find the Lambert W solution of the following equation, but I am not able to proceed further because I am not able to put the equation in the form of $x=W(x)\exp(W(x))$.
The equation is: $$d = x(1+x) + \ln(1+x)$$ where $d$ is constant.
I am looking for real solutions.
On
Lagrange reversion gives:
$$f(x)=x(x+1)+\ln(x+1)\implies f^{-1}(x)=\sum_{n=1}^\infty\frac{(-1)^n}{n!}\frac{d^{n-1} e^x(e^{2x}-e^x)^n}{dx^{n-1}}$$
Expand with the derivative’s argument with the binomial theorem and get:
$$\boxed{f^{-1}(x)=e^x-1+\sum_{n=1}^\infty\sum_{m=0}^n\frac{(-1)^m(m+n+1)^{n-1}e^{(m+n+1)x}}{(n-m)!m!}}$$
shown here when comparing links. It works for $x\le 0$:
$$d=x(1+x)+\ln(1+x)$$ $$\ln(1+x)+x^2+x-d=0$$
We see, the equation is a polynomial equation of more than one algebraically independent monomials ($x,\ln(1+x)$) and with no univariate factor. We therefore don't know how to rearrange the equation for $x$ by applying only finite numbers of elementary functions (operations) we can read from the equation.
a)
$$d=x(1+x)+\ln(1+x)$$
Let's exponentiate both sides of the equation, as proposed by one of the comments.
$$e^d=(1+x)e^{x(1+x)}$$
We see, this equation is not in a form for applying Lambert W or Generalized Lambert W.
b)
$$d=x(1+x)+\ln(1+x)$$ $x\to e^t-1$: $$d=(e^t-1)e^t+\ln(e^t)$$ i.a. for $t\in\mathbb{R}$: $$d=\left(e^t\right)^2-e^t+t\tag{1}$$
We see, this equation is not in a form for applying Lambert W or Generalized Lambert W.
We see, equation (1) is an irreducible algebraic equation of $t$ and $e^t$ simultaneously if $d$ is an algebraic number.
According to the theorems in [Lin 1983] and [Chow 1999], such kind of equations cannot have solutions except $0$ that are elementary numbers or explicit elementary numbers respectively. And $x$ can therefore also not be an elementary number except $0$.
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[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Mező 2017] Mező, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553
[Mező/Baricz 2017] Mező, I.; Baricz, Á.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)
[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018