How would you solve it -Part 2

Calculus Level 5

\[ \large \displaystyle \int_0^{\infty}\left(\sqrt{1+x^4} -x^2 \right) \mathrm{d}x = \dfrac{\left(\Gamma\left(\frac{1}{b}\right) \right)^a}{c\pi^d} \]

If the above equation holds true for positive integers \(a,b,c,\),and a real number \(d\) where \(\Gamma\) is the Gamma function, find the value of \((a+b+c+2d)^2\).

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