Maclaurin series of an irrational fraction

Question

I can't figure out what's the proper series expansion of the function $$f(x)=\frac{(1+2x)\sin(x)}{\sqrt{1+x^2}}$$ at least to the third degree. WolframAlpha gives a quite clean solution, and the best I can come up is calculating by definition, which gets really messy at the third derivative (2nd is ugly but doable). I was wondering if there's a smarter way?

Answer 1

Hint: Start by writing $f(x)=(1+2x)\cdot(\sin x)\cdot (\frac{1}{\sqrt{1+x^2}})$. Now, for each term, write out the Maclaurin series (the third one is, perhaps, a little trickier but not difficult), just taking the first couple of terms in each Maclaurin series. Then you can just expand by multiplying.