A problem by Refaat M. Sayed

$\int\limits^{\infty}_{0}\frac{1}{(x^8+5x^6+14x^4+5x^2+1)^{4}}dx =\pi \frac{14325195794+\left( A\sqrt{B} \right) }{C\left( D+E\sqrt{F} \right) ^{\frac{G}{H} }}$

$A,B,C,D,E,F,G,H$ are positive integers with $B,F$ square-free, both $(D,E)$ and $(G,H)$ are coprime pairs and $C$ is minimized.

Find $A+B+D+E+F+G+H+E+F+G$ .