Who's up to the challenge? 72

Calculus Level 5

\[\large \int _{ 0 }^{ 1 }{ { x }^{ 3 }\sqrt { 1-{ x }^{ 7 } } dx } =\frac { A\Gamma \left( \frac { B }{ C } \right) \sqrt { \pi } }{ D\Gamma \left( \frac { E }{ F } \right) } \]

If the equation above holds true for positive integers \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) with \(\text{gcd}(A,D)\)\(=\text{gcd}(B,C)\)\(=\text{gcd}(E,F)=1\) and \(\dfrac{B}{C},\dfrac{E}{F}<1\), find \(A+B+C+D+E+F\).

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