Would IBP work?

Calculus Level 5

Evaluate: 0x3ln2x1+x6dx\int_0^{\infty} \frac{x^3\ln^2 x}{1+x^6}\,dx

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

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