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I've come upon an integral of an exponential raised to a ration fraction in the form:

$$ \int_0^{\infty} e^{- (x^2 + ax + b + c/x)} dx $$

All constants $a,b,c > 0$. I'm convinced that it converges.

I'm curious if there's any analytic expression for this definite integral. Mathematica gives no solution.

Similar forms like How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$ and $f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$? use substitution tricks/completing the square to leverage the inverse power of $x$, which do not work (as far as I've tried) in this specific case.

Thank you.

metamorphy
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kdmeaney
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    The tricks those questions use seem to more or less rely on the symmetry inherent to those integrals and basically devolve into clever use of $(x+1/x)^2$. I don't think those methods would work here; something better is needed. – PrincessEev Jan 27 '22 at 23:20
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    may be partial differerentiation by some parameter can help. – Sam Ginrich Jan 28 '22 at 12:37
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    I can get it to $I = \frac{e^{-b}}{4} \int_{0}^{\infty} J_0 \left(0, 2\sqrt{c x} \right) \left(2-\sqrt{\pi} (a+x) e^{\frac{(a+x)^2}{4}} , \text{erfc} \left(\frac{a+x}{2}\right) \right) , dx$ Where $J$ is the Bessel J function. The $\frac{c}{x}$ really complicates it. – KStar Jan 28 '22 at 13:58

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