general procedure for contour integration of $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$

Question

In Richard Crandall's On The Quantum Zeta Function, following eq. 4.11:

$$ \int_{0}^{\infty}\mathrm{Ai}(x)^{2}dx=\frac{\Gamma(\tfrac{5}{6})}{2\pi^{5/6}12^{1/6}} $$

“again derivable by contour integration”

• Which contour? (for this)
• Can the contour be generalized for for $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$?