How to solve this integral $\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$

Question

My classmate asked me about this integral:$$\int_0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$$ but I have no idea how to do it. What's the closed form of it? I guess it may be related to the Airy function.

Answer 1

I guess it may be related to the Airy function.

You guessed well. In general, we have $~\displaystyle\int_0^\infty\exp\Big(-x^2(x+3a)\Big)~dx~=~\frac2{e^{2a^3}}\cdot\frac{\text{Bi}\Big(3^{2/3}~a^2\Big)}{3^{4/3}}-$

$-a\cdot~_2F_2\bigg(\bigg[\dfrac12~,~1\bigg]~;~\bigg[\dfrac23~,~\dfrac43\bigg]~;~-4a^3\bigg).~$ In this particular case, $a=-\dfrac23.$

Answer 2

Apparently there is a closed form that includes the generalized hypergeometric functions. They themselves can be expressed in a series, so you might be able to trace back the solution to some kind of series expression of the integrand:

http://www.wolframalpha.com/input/?i=Integrate+Exp[-x^3%2B2+x^2+%2B1]+from+0+to+Infinity

My approach would be to take the solution and write the functions as a hypergeometric series.