definite integral of Gaussian function over polynomial of even powers

Question

I need to know if there is a unified solution in terms of $L$ and $a_i>0, i=0,...,L$ for following integral: $\int_{0}^{\infty} \frac{e^{-x^2}}{\sum_{i=0}^{L} a_ix^{2i}}dx$. If not, what if we specify $L$, e.g. $L=4$. Can we solve it for $L=4$? Thanks for any help in advance.

Answer 1

For an arbitrary polynomial, chances of a closed-form expression are nil.

Because there will be no closed-form for the indefinite integral (think that even a pure Gaussian requires a special function), and a definite integral with parameters is tantamount to an indefinite one.

Indeed, as the denominator is an arbitrary polynomial, by a shift of the variable you get another polynomial, hence the lower bound being $0$ is immaterial.

Alpha gives up for a simple case such as with $L=1$,

$$\int\frac{e^{-x^2}}{1+x^2}dx.$$

https://www.wolframalpha.com/input/?i=integrate+e%5E(-x%5E2)%2F(x%5E2%2B1)

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I am not even referring to the fact that $L=4$ yields a quintic polynomial in $x^2$, for which it has since long been proven that no closed-from expression exists. (After extremely arduous computation, you can reduce one to the Bring-Jerrard form, $x^5-x+y=0$, requiring an ad-hoc transcendental function.)