Are d orbitals included in hybrid orbitals?

According to Linus Pauling and this reference, the answer is "sometimes"

http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/hybrid.html

I once audited a course taught by Pauling at Caltech. I also bought his famous book "The Nature of The Chemical Bond". I didn't get far in either. It was all Greek to me.

Yes, d orbitals can be included in hybrid orbitals depending on the atom's configuration and the nature of the bonding in the molecule.

When atoms form covalent bonds, they can undergo hybridization to form hybrid orbitals. Hybrid orbitals are a combination of atomic orbitals that have different shapes and energies. The most common types of hybrid orbitals are sp, sp2, and sp3, which involve the mixing of s and p orbitals. However, in certain cases, d orbitals might also be involved in hybridization.

For example, in molecules or ions with an expanded octet, such as SF6 or PCl5, the central atom has more than four bonded atoms or electron pairs, and can therefore involve d orbitals in hybridization. In these cases, the central atom's s, p, and d orbitals can mix to form hybrid orbitals, such as sp3d or sp3d2.

To determine whether d orbitals are included in hybrid orbitals, you need to examine the electron configuration and the geometry of the molecule or ion. By studying the Lewis structure, VSEPR theory, and the octet rule, you can determine the hybridization of the central atom and identify the types of orbitals involved.