To calculate bending moment [M] on a mixer shaft, you need to compute the bending moment at each impeller location.  The impeller location or shaft length [L] to each impeller is measured from the bottom support bearing of the mixer drive. 
You describe a 70:30 power split between the two impellers, make that split based on the motor horsepower [Hp]. You want to account for the maximum load on each impeller.  In addition to the shaft length [L] and power split [Hp], you need to know the rotational speed [N] of the mixer and the impeller diameter [D] of each impeller, then apply the following formula for each impeller and sum the results:
 
M (in-lbs) = 19000 Hp (fraction motor horsepower) × hydraulic factor × L (inches) / N (rpm) / D (inches)
 
The hydraulic factor should be 3.0 for the gas dispersion impeller, because of fluctuating loads caused by the gas.  The upper impeller should also have a hydraulic factor of 2.0 because of the gas or 3.0 if it must operate at the liquid level during filling or emptying of the tank.

The bending moment calculation will not complete a shaft analysis.  The shaft torque must be calculated and then tensile and shear stresses can be estimated, which will reflect the strength of the mixer shaft.  A natural frequency calculation is also appropriate to avoid undamped vibrations.