For what it's worth, the Cauchy principal value of the integral for $B=0$ has a lengthy expression in terms of Meijer G-functions,
$$-8 \pi ^{5/2} \int\limits_{-\infty}^{\infty} \frac{e^{i A x^2}}{x - a} \, dx=G_{3,5}^{5,3}\left(\alpha\,\biggl|
\begin{array}{c}
 0,\frac{1}{4},\frac{3}{4} \\
 0,0,\frac{1}{4},\frac{1}{2},\frac{3}{4} \\
\end{array}
\right)+8 \pi ^4 G_{7,9}^{5,3}\left(\alpha\,\biggl|
\begin{array}{c}
 0,\frac{1}{4},\frac{3}{4},-\frac{1}{8},\frac{1}{8},\frac{3}{8},\frac{5}{8} \\
 0,0,\frac{1}{4},\frac{1}{2},\frac{3}{4},-\frac{1}{8},\frac{1}{8},\frac{3}{8},\frac{5}{8} \\
\end{array}
\right)+i G_{3,5}^{5,3}\left(\alpha\,\biggl|
\begin{array}{c}
 \frac{1}{4},\frac{1}{2},\frac{3}{4} \\
 0,\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4} \\
\end{array}
\right)+8 \pi ^4 i G_{7,9}^{5,3}\left(\alpha\,\biggl|
\begin{array}{c}
 \frac{1}{4},\frac{1}{2},\frac{3}{4},-\frac{1}{8},\frac{1}{8},\frac{3}{8},\frac{5}{8} \\
 0,\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},-\frac{1}{8},\frac{1}{8},\frac{3}{8},\frac{5}{8} \\
\end{array}
\right),$$
with $\alpha=a^4A^2/4$.