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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