Bounding integral with quadratic complex exponential

Question

I am reading this book (Euclidean Harmonic Analysis, Benedetto, 1979) and on pages 24-25 Carleson proves a lemma related to the Kolmogorov-Seliverstov-Plessner method. There is one small step in bounding a Fourier-transform-esque integral that he takes as trivial, but I cannot seem to figure out how he came to the conclusion. Namely, on the first line of p. 25 he states that if $|A|\leq 2$ and we take $0<\alpha<1$ then the integral $$\int_{-\infty}^\infty e^{i(2At+t^2)}\frac{dt}{|t|^{\alpha}}$$ is bounded. This seems logical, but I am unsure what the standard approach to bounding such integrals is (do we use Contour integration? Some sort of analytic argument involving cancellation?). Many thanks in advance.