Who's up to the challenge?10

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

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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