Why don't helicopters prefer shorter rotors with more blades? My understanding of helicopters is that, despite theoretically increasing efficiency, longer blades are worse in practice than short ones in every respect (except, perhaps, cost):


Longer blades are more prone to vibration and resonance dynamics, if for no reason other than that their rigidity is reduced by length, holding all else constant.
Longer blades provide less operating clearance.
Longer blades not only stall before shorter blades (retreating), but also hit the Critical Mach number at lower air speeds (advancing).
Fewer blades require higher rotational speeds for a given "rotor disk" area, which is inefficient from not only an aerodynamics perspective but also, presumably, a transmission-and-control perspective.


But all helicopters have few blades of significant length.  So: What are the advantages of longer blades, or disadvantages of shorter blades, that lead to the rotor designs in common use?

I would imagine that in the limit it would be optimal to run something more like a ducted turbine segment, where the swept area is virtually covered in blade surface (i.e., maximizing the number of blades over the swept disc).

Noting that in practice blade size seems somewhat proportional to the body size of the aircraft, I'll deduce that in practice the bulk of the downforce needs to be generated some minimal distance away from the aircraft's body, though with some attention to the body's vertical aerodynamic aspect it seems like that could be reduced.  I guess tilt-rotors could support this hypothesis since their rotors are positioned at a distance from the body and they use smaller rotors, although it seems like they still use blades as large as possible given the wing size and body configuration.
The rotor works by accelerating air downwards, therefore creating an upward reaction force on the blades that lifts the craft.

The lift force is equal to

$$L = \dot{m}\Delta v$$

where $\dot{m}$ is mass flow rate through the rotor and $\Delta v$ is the change of speed of the air. To accelerate the air to that speed, it has to give it kinetic energy. This requires power

$$D = \frac12\dot{m}\Delta v^2$$

(the induced drag).

Now a rotor with smaller diameter will be able to affect less air, so the $\dot{m}$ will be lower. To produce the same lift, it will need larger $\Delta v$ and therefore it will need more powerful engine (up to a point, because longer blades will have higher form drag that will eventually outweigh the reduction in induced drag) and, more importantly, burn more fuel.

The blades, like wings, affect air to a significant distance above and below them, so you don't need many blades to use all air within the rotor area. But no matter how many blades you add, they will still only affect the air within the rotor area, so the efficiency won't change much.

As far as tiltrotors go, they have relatively small rotors and that makes them awfully inefficient in hover. But they don't spend most of their time hovering, they spend most time flying somewhere and in the fixed-wing mode they are faster and more efficient, which outweighs the worse efficiency during take-off and landing.

See also: Is there any equation to bind velocity, thrust and power? (via Energizer777's deleted answer).