I would like a simple, computationally viable expansion in $x$ of $\sqrt{1+f[x]}$. If we taylor expand $f[x] $ itself inside the square root to first couple of orders, and expand the square root for small x we have for example,
$\sqrt{1+f[0]+\sum_{n=1}^{\infty}\frac{\partial^{n}_xf[x]|_{x=0}}{n!}x^n}$= $\frac{x f'(0)}{2 \sqrt{f(0)+1}}+\frac{x^2 \left(2 f(0) f''(0)+2 f''(0)-f'(0)^2\right)}{8 (f(0)+1)^{3/2}}+\sqrt{f(0)+1}$.
Does this type of expression has a closed form? I suspect one can rewrite this as exp of log and then use the Faa di Bruno formula for the exponential, but this seems to require expanding $\log{\left(1+f[0]+\sum_{n=1}^{\infty}\partial^{n}_xf[x]|_{x=0}x^n\right)}$, which also seems complicated.