irrational to integer

$\begin{aligned} D & = \sqrt{12-\sqrt{24}+\sqrt{39}-\sqrt{104}} - \sqrt{12+\sqrt{24}+\sqrt{39}+\sqrt{104}} & \small \color{#3D99F6} \text{Let }a = 12 + \sqrt{39} \text{ and }b = \sqrt{24} + \sqrt{104} \\ & = \sqrt{a-b} - \sqrt{a+b} \\ \implies D^2 & = (a-b) - 2\sqrt{a^2 - b^2} + (a+b) \\ & = 2a - 2\sqrt{a^2-b^2} \\ & = 24 + 2\sqrt{39} - 2\sqrt{183+24\sqrt{39} - 128 - 16\sqrt{39}} \\ & = 24 + 2\sqrt{39} - 2\sqrt{55+8\sqrt{39}} \\ & = 24 + 2\sqrt{39} - 2\sqrt{(4+\sqrt{39})^2} \\ & = 24 + 2\sqrt{39} - 8 - 2\sqrt{39} \\ & = 16 \\ \implies D & = \boxed{- 4} & \small \color{#3D99F6} \text{Note that }D < 0 \end{aligned}$