$d$-dimensional Fourier integral of log term

Question

I was wondering if there exists a general formula for Fourier transform integrals of this type, which appear frequently in qft

$$
\int \frac{d^d k}{(2\pi)^d} \, e^{i\vec{k}\cdot \vec{x}} [\log k^2]^n [k^2]^m,
$$ 
where the metric is euclidean.
I know a general formula that holds in dimensional regularization for the same integrals without the log term, or for the integral with $n=1, m=0$, but I cannot find anything similar for the integral above.. Does anyone know if these integrals can be regularized and have a closed expression? Even something for a fixed dimension $d$ would be cool.

Thank you so much!