Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)

Theorem

 * $\ds \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$

where $a$ and $b$ are strictly positive real numbers.