Infinitive forms are used with all three of these verbs; the difference is that causative HAVE and MAKE take ‘bare’ infinitives, while GET takes infinitives ‘marked’ with the particle to.
The reason for the difference is historical. Very briefly, and ignoring a lot of fine detail: the idioms with MAKE and HAVE are very old, and became current at a time before the use of to infinitives was fully developed. Causative GET, however, is a fairly recent development, not attested until the 15th century, so it follows the ‘modern’ practice of chaining verbs with to infinitives.
Causative MAKE and HAVE are acceptable in all registers, but causative GET is mostly conversational; it is rare in formal registers. There is some distinction in sense:
GET and HAVE are merely causative; in your example, for instance, they imply only that your brother carried your suitcase because you induced him to.
MAKE, however, implies compulsion: in your example it implies that your brother was unwilling to carry your suitcase and would not have carried it without some insistence or threat on your part.