A Radical Integral

We can solve this integral by making the substitution $u=5+\sqrt{x}\,.$ Then,

$(u-5)^2 = x \,\,\, \Rightarrow\,\,\,\mathrm{d}x = 2(u-5)\, \mathrm{d}u\,.\,\,$ So our new intergal is

$\begin{aligned} \displaystyle\int_5^9 2\sqrt{u}\,(u-5)\,\mathrm{d}u &=\displaystyle\int_5^9 \left(2\,u^{3/2} - 10\,u^{1/2}\right)\,\mathrm{d}u \\ &=\displaystyle\left[\dfrac{4}{5}\,u^{5/2}-\dfrac{20}{3}\,u^{3/2}\right]_5^9 \\ &=\displaystyle \dfrac{72}{5} + \frac{40\sqrt{5}}{3} \\ &=\displaystyle \dfrac{216+200\sqrt{5}}{15} \end{aligned}$

So, $\displaystyle A+B+C+D=\boxed{436}\,.$

Great problem dude! :D