Symmetric Rearrangement

Calculus Level 5

\[\int _{ 0 }^{ 1 }{ { \bigg [ \left( 1-{ x }^{ 1/17} \right) }^{ 23 } } +{ \left( 1-{ x }^{ 1/23 } \right) }^{ 17 } \bigg ] dx\]

The integral above is in the form \(A \frac{B!C!}{40!} \) for positive integers \(A, B, C \).

Calculate the smallest value of \(A \times B \times C \times 40 \).

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