Stone-Geary utility function, derivation of Marshallian demand
If I understand your question correctly, you can take the log of the original utility function to simplify the calculations.  The demand derived will be the same as that which is derived from the original utility function.  This follows since the log function is a monotone function, and preferences are preserved under monotonic transformations of utility.  Thus, the "log-utility" case that you find online is exactly what you want to work with since it would simplify things quite a bit.