I need the series expansion of a fairly nasty function and its derivative:
f = (x - a) (1 + b (1/(x - a))^((5 + Sqrt[73])/12) *
Hypergeometric2F1[(-7 + Sqrt[73])/12, (1 + Sqrt[73])/12, (
6 + Sqrt[73])/6, -(a/4)/(x - a)]) + a/2
$f\left(x\right) = (x-a) \left(1+b \left(\frac{1}{x-a}\right)^{\frac{1}{12} \left(5+\sqrt{73}\right)} \, _2F_1\left(\frac{-7+\sqrt{73}}{12} ,\frac{1+\sqrt{73}}{12} ;\frac{6+\sqrt{73}}{6} ;-\frac{a}{4 (x-a)}\right)\right)+\frac{a}{2}$
where $x$, $a$ and $b$ are positive real numbers and I know $x \gg a$. I'm interested in the series expansion of $f$ and $f'$ in inverse powers of $x$:
Series[f, {x, \[Infinity], 1}] // Normal
$\dfrac{b \left(\dfrac{1}{x-a}\right)^{\dfrac{\sqrt{73}}{12}}}{\left(\dfrac{1}{x}\right)^{7/12}}-\dfrac{a}{2}+x$
Series[D[f, x], {x, \[Infinity], 1}] // Normal
$b \left(\dfrac{1}{x}\right)^{5/12} \left(\dfrac{1}{x-a}\right)^{\dfrac{\sqrt{73}}{12}}+1$
The problem is I can't figure out how to get this to simplify any further. I need to get these in the form
$ c_1 x + c_0 + \dfrac{c_{-1}}{x^{p}} $
for some constants $c_i$ and $p$, but Series
doesn't expand the term with $x-a$ in the denominator. I guess because the power is irrational? How can I get the answer in the desired form? I could do it by pen and paper of course, but I would love to have it in code. Using Assumptions -> {x > a > 0, b > 0}
doesn't seem to make any difference.
Any help is appreciated.