OMG !! Zeta and Pi in one integral!

Calculus Level 5

If 011x1x6(lnx)4 dx=aπkbc+dζ(5)e\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,ka,b,c,d,e,k are positive integers and cc is not divisible by any perfect square. Find a+b+c+d+e+ka+b+c+d+e+k


Details and assumptions : ζ(5)=k=11k5.\zeta(5) = \sum_{k=1}^{\infty} \frac{1}{k^5} .

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