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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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