OMG !! Zeta and Pi in one integral!

Calculus Level 5

If \[\int_0^1 \frac{1-x}{1-x^6}(\ln x)^4 \ \mathrm{d}x= \frac{a\pi^k}{b\sqrt{c}} + \frac{d\zeta(5)}{e} \] Where \(a,b,c,d,e,k\) are positive integers and \(c\) is not divisible by any perfect square. Find \(a+b+c+d+e+k\)


Details and assumptions : \[\zeta(5) = \sum_{k=1}^{\infty} \frac{1}{k^5} .\]

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