Highschool tricks

Calculus Level 3

\[ \large \int_0^\infty x^2 e^{-\left( x^2 + \left(\frac{3}{2x}\right)^2 \right)} \, dx \]

If the integral above equals to \( \frac{ \sqrt[a]{\pi} }{e^b} \) for integers, \(a,b\), find the value of \(a+b\).

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