\[\large{\displaystyle \int^{\infty}_{0} \frac{1}{6x^3+7} \left( \frac { { d }^{ \frac { 1 }{ 2 } }x }{ d{ x }^{ \frac { 1 }{ 2 } } } \right) dx}\]

The value of above expression equals \(\large{\frac{\pi^{A}}{B^{C}D^{E}F^{G}}}\)

for prime numbers \(B,D,F\) and \(A,C,E,G\in \mathbb R\)

Find \(A+B+C+D+E+F+G\).

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