I would like to obtain the Taylor expansion of $\color{red}{g(x)=\left(x+a-\sqrt{x(x+2a)}\right)^{\alpha}}$ at $\color{blue}{x=x_0>0}$. Here, $a>0$ and $\alpha\in\mathbb{R}$
Note that
$$\left(x+a-\sqrt{x(x+2a)}\right)^{\alpha}=\left(x+a-\sqrt{(x+a)^2-a^2}\right)^{\alpha}=\left(\frac{\sqrt{x+2a}-\sqrt{x}}{2}\right)^{2\alpha}$$
Since calculate the n$th$ derivative of $g(x)$ is complicated, I wonder if is possible to get a closed form for this n$th$ derivative (I don't know) or use a known series of another related function (Which? How?).
Any help will be welcomed