TITLE: The Apéry's constant and the Airy function
QUESTION [9 upvotes]: Several weeks ago, while I was playing with a CAS, Wolfram Alpha online calculator I found the closed-form that provided this calculator for $$\int_0^\infty\operatorname{Ai}(x)\log^3(x)dx,\tag{1}$$
involving the Airy function $\operatorname{Ai}(x)$, see this special function from the Wikipedia Airy function.
But I don't know nor if it is in the literature neither nor any hint to get the corresponding indefinite integral (if it is feasible). 

Question. Do you know if the closed-form for $$\int_0^\infty\operatorname{Ai}(x)(\log(x))^3dx,\tag{1}$$ in terms of well-known constants and $\zeta(3)$ is in the literature? Provide the reference, and if I can I search it. In other case, can you provide an hint to calculate this kind of definite integral? Many thanks.

REPLY [2 votes]: Using the various provided answers, I present to you the craziest integral I've ever seen. Behold: 

$$\int_0^\infty \mathrm{Ai}(x)\ln^3x\ dx=-\frac1{81}\left[(4\pi^2+\beta^2)\beta+52\zeta(3)\right]$$
  where $\beta=2\gamma+\ln3$.