$\quad \sqrt{a^2 + b^2 + \dfrac{a^2}{\left(\dfrac ab +1\right)^2}}\qquad (b\ne 0)$
$= \sqrt{a^2 + 2ab + b^2 - 2ab + \dfrac{a^2}{\left(\dfrac ab +1\right)^2}}$
$= \sqrt{(a+b)^2 - 2\cdot\dfrac{ab(a+b)}{a+b} + \dfrac{a^2}{\left(\dfrac ab +1\right)^2}}$
$= \sqrt{(a+b)^2 - 2\cdot\dfrac{a(a+b)}{\dfrac{a+b}{b}} + \dfrac{a^2}{\left(\dfrac ab +1\right)^2}}$
$= \sqrt{(a+b)^2 - 2(a+b)\cdot\dfrac{a}{\dfrac{a}{b}+1} + \dfrac{a^2}{\left(\dfrac ab +1\right)^2}}$
$=\sqrt{\left(a+b -\dfrac{a}{\dfrac ab + 1}\right)^2}$
$= \left|a+b -\dfrac{a}{\dfrac ab + 1}\right|$
