Evaluating the Fourier transform of the following piecewise functions

Question

I'm looking to evaluate the Fourier transform of the piecewise functions, $f(y)$ and $g(y)$ consisting of a Bessel function with square root argument, a Gaussian and a linear function/algebraic function:

1. \begin{equation} f(y)=\begin{cases} J_0(b\sqrt{a^2-y^2})(y-p)e^{\frac{-(y-c)^2}{2d^2}}\quad\text{for $0a$} \end{cases} \end{equation}

2. \begin{equation} g(y)=\begin{cases} \frac{J_1(b\sqrt{a^2-y^2})}{\sqrt{a^2-y^2}}ye^{\frac{-(y-c)^2}{2d^2}}\quad\text{for $0a$} \end{cases} \end{equation} where $b>0$ , $c>0$ , $d>0$ and $Re[p]>0$

I tried this by changing $\sqrt{a^2-y^2} \rightarrow u$ and then applying integration by parts. However, the integral becomes cumbersome due to the presence of a Gaussian and a algebraic function.

I'm using the definition, $H(k)=\int_{-\infty}^{+\infty} h(y)e^{iky}dy$ for the Fourier transform.

Any suggestions on further evaluation would be much appreciated.