# Is there any relationship to the Gaussian integral?

Calculus Level 4

The Gaussian integral states that $$\displaystyle \int_{-\infty}^{+\infty} e^{-x^2 } \, dx = \sqrt{\pi }$$.

Given that information (if it is related to the problem at hand), evaluate

$\int_{- \infty}^{+ \infty} x^{2} e^{-x^4} \, dx$

If the integral above can be represented in the form

$\dfrac{a}{b} \Gamma \left( \dfrac{c}{d}\right) \; ,$

where $$a,b,c$$ and $$d$$ are positive integers such that $$\gcd(a,b)=\gcd(c,d) =1$$, find $$a+b+c+d$$.

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