Taylor series in Maple for root of $x=e^{a\frac{\ln(1+b/x)}{\ln(1+b/x)+c}}$

Question

I want to find the Taylor series for the following function in Maple:

Root of $$x=\large e^{a\cdot \dfrac{\ln\left(1+\frac{b}{x}\right)}{\ln\left(1+\frac{b}{x}\right)+c}}$$ when $a,b,c>0$

I use: series(RootOf(m = exp(a*ln(1+b/x)/(ln(1+b/x)+c)), m), a)

Maple returns:

\begin{align} \label{eq:1} 1+\dfrac{\ln\left(\dfrac{b+x}{x}\right) }{\ln\left(\dfrac{b+x}{x}\right)+c}a+\dfrac{\ln\left(\dfrac{b+x}{x}\right)^2 }{2\left(\ln\left(\dfrac{b+x}{x}\right)+c\right)^2}a^2+\cdots+O(a^6) \end{align}

[fourth to sixth terms are dropped for brevity]

As you see, the parameter $x$ appears again which is not what we want. Do you know how I can solve this?

If MAPLE does not work, let me know if you can do this in MATLAB or MATHEMATICA

Second question, do you know how I can convert the output style of MAPLE to Latex? The output I got from MAPLE was originally different (see below) and it took me some time to convert it to a beautiful form written above.

1+a*ln((b+x)/x)/(ln((b+x)/x)+c)+(1/2)*ln((b+x)/x)^2*a^2/(ln((b+x)/x)+c)^2+(1/6)*ln((b+x)/x)^3*a^3/(ln((b+x)/x)+c)^3+(1/24)*ln((b+x)/x)^4*a^4/(ln((b+x)/x)+c)^4+(1/120)*ln((b+x)/x)^5*a^5/(ln((b+x)/x)+c)^5+O(a^6)

Answer 1

I am not sure why you changed the x to and m, it should be:

series(RootOf(x = exp(a*ln(1+b/x)/(ln(1+b/x)+c)), x), a);

You can use the latex() command to produce latex markup of a Maple expression. I got the following from:

latex(series(RootOf(x = exp(a*ln(1+b/x)/(ln(1+b/x)+c)), x), a, 2));

$$1+{\frac {\ln \left( b+1 \right) }{\ln \left( b+1 \right) +c}}a+{ \frac {\ln \left( b+1 \right) \left( \left( \ln \left( b+1 \right) \right) ^{2}b+\ln \left( b+1 \right) bc+ \left( \ln \left( b+1 \right) \right) ^{2}+\ln \left( b+1 \right) c-2\,bc \right) }{2\, \left( \ln \left( b+1 \right) +c \right) ^{3} \left( b +1 \right) }}{a}^{2}+O \left( {a}^{3} \right)$$