Evaluate: \[\int_0^{\infty} \frac{x^3\ln^2 x}{1+x^6}\,dx \]

If the result can be expressed as \(\displaystyle \frac{a\pi^b}{c\sqrt{d}}\), where \(a\) and \(c\) are coprime, \(b\) is an integer and \(d\) is a prime number, then find \(a+b+c+d\).

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