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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