moments (and/or density) for "std" in the "rugarch"-package or "TF2" in the "gamlss.dist"-package

Dear R community,

in order to get a formula for the kurtosis of a mixture distribution 
with noncentral scaled student t components I need to get some raw 
moments of the components (which will then be applied to the law of 
total expectation).

As I am working with the noncentral scaled student t implementations 
"std" in the "rugarch"-package and "TF2" in the "gamlss.dist"-package I 
would be happy to get more details about the particular implementation 
of the distributions.
As to my knowledge the two mentioned implementations are equivalent.
Importantly, the scaling factor equals the standard deviation.

The particular density of the noncentral scaled student t distributions 
as implemented in those packages would be helping.
And if there exist nice formulas for the (noncentral or central) moments 
up to order 4 for this distribution I would be more than thankful for 
any link or comment!

Best,
Johannes