How to approximate the following function? $$f(x) = \frac{a_1x^2+a_2x+a_3}{\sqrt{a_4x^2+a_5x+a_6}} + \frac{b_1x^2+b_2x+b_3}{\sqrt{b_4x^2+b_5x+b_6}}$$ where $a_i$ and $b_i$ are constants.
I thought about compute $f(x)^2$ and then expand the square but didn't find it fruitful. My intuition is $$f(x) \approx \frac{c_1x^2+c_2x+c_3}{\sqrt{c_4x^2+c_5x+c_6}}$$ for some $c_i$ but didn't find the right tool to prove it.