Simplify $\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$

Question

I would like to rewrite this integral

$$\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$$ (where $a>\mathbf{R^+}$ and $J_{\frac{3}{2}}$ is the bessel function of the first kind of order $3/2$)

in such away I have something of the form:

$$\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}=1-\int_0^{\infty}\,dk\,f(k)$$

where $f(k)$ is a generic function in $k$.

I tried by considering $k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)$ a first derivative in $k$ of a function and then using the integration rule by parts...but I don't end up with the solution I seek.

Any help? Thanks

Answer 1

Using Maple I am obtaining that the integral is exactly evaluated as