Simplify: $\;$ $\dfrac{\left(\dfrac{\sqrt[4]{bx^3} + \sqrt[4]{a^2 bx}}{\sqrt{x} + \sqrt{a}} + \sqrt[4]{bx}\right)^2 + bx + 4}{x \left(\sqrt{b} + \sqrt{4 x^{-1}}\right)^2}$
$\dfrac{\left(\dfrac{\sqrt[4]{bx^3} + \sqrt[4]{a^2 bx}}{\sqrt{x} + \sqrt{a}} + \sqrt[4]{bx}\right)^2 + bx + 4}{x \left(\sqrt{b} + \sqrt{4 x^{-1}}\right)^2}$
$= \dfrac{\left(\dfrac{b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}} + b^{\frac{1}{4}} x^{\frac{1}{4}}\right)^2 + bx + 4}{x \left(\sqrt{b} + \dfrac{2}{\sqrt{x}}\right)^2}$
$= \dfrac{\left(\dfrac{b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}} + b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$
$= \dfrac{\left(\dfrac{2 b^{\frac{1}{4}} x^{\frac{3}{4}} + 2 a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$
$= \dfrac{\left(\dfrac{2 b^{\frac{1}{4}} x^{\frac{1}{4}} \left(x^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$
$= \dfrac{4 b^{\frac{1}{2}} x^{\frac{1}{2}} + bx + 4}{bx + 4 + 4 b^{\frac{1}{2}} x^{\frac{1}{2}}}$
$= 1$