I have a trouble giving an equivalent of this integral (and of course, Maple is unable to give it). Let $ \mathrm{Ai} $ denote the Airy function whose zeroes are denoted by $ 0 > \alpha_1 > \alpha_2 > \dots $ (see wikipedia for a plot). I form the continuous function $ f_k $ defined by \begin{align*}%$ f_k(x) := \frac{ \mathrm{Ai}(x) }{x - \alpha_k} \quad (x \neq \alpha_k), \qquad f_k(\alpha_k) = \mathrm{Ai}'(\alpha_k) \end{align*}
Question : I would like an equivalent of \begin{align*}%$ g(x) := \int_{-\infty}^{-x} f_k(y)^2 dy \end{align*} when $ x \to +\infty $.
Note that this last integral is well-defined due to the decay of $ x\mapsto \frac{1}{(x - \alpha_k)^2} $.
I thought about rescaling $ g(x) = x \int_{-\infty}^{-1} f_k(xt)^2 dt $ and use the equivalent of $ \mathrm{Ai} $ inside the integral, but the equivalent I get is not integrable...