Who's up to the challenge?10

Calculus Level 5

\[ \displaystyle\int _{ 0 }^{ \infty }{ \left( ({ x }^{ 5 }+1)^{ \frac { 1 }{ 5 } }-x \right)} \, dx =\dfrac { A }{ C } B\left(\dfrac { D }{ F } ,\frac { 1 }{ G } \right)\\ \]

Given that the equation holds true where \(B(x,y)\) is the beta function, and \(A,C,D,F,G\) positive integers \(\gcd(A,C) = \gcd(D,F) = 1\). Find \(\text{min}\{A+C+D+F+G\}\)


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