Taylor series expansion of $\frac{(1+x)}{1}+\frac{(1+x)^\frac{1}{3}}{3}+\frac{(1+x)^\frac{1}{5}}{5}+\cdots$

Question

I wanted to find the Taylor expansion of the summation:

$\dfrac{(1+x)}{1}+\dfrac{(1+x)^{1/3}}{3}+\dfrac{(1+x)^{1/5}}{5}+\cdots$ .

I can't evaluate it normally using the first derivative, second derivative... method since the summation terms don't equal $0$ when $x=0$

So I broke down the terms individually, $(1+x)^\frac{1}{3}=a_0+a_1x^1+\cdots=$ and $(1+x)^\frac{1}{5}=b_0+b_1x^1+\cdots$

However this procedure is tedious and requires a lot of summations for each $x^n$ term. Is there a better way of evaluating the Taylor series?