Well, Mathematica does a very general check of the expression to find any part that it can simplify. So if your expression contains too many radicals and Log function, it would be fairly difficult to handle since you might not be aware of the really complicated conditions which requires too much computation. A good idea is always to simplify this type of problem at first step.
At first glance the up limit of the integral can be simplified. They do not have dependance. I will just go ahead the do the integral separatedly:Then I will have a result. Use Rule and Replace functions to find the value of the first step integral (I just copy and paste the results): 

Then you can copy the above two expressions into the new integrate function:The result is

Finally, if you really want to have the up limits to be your version, you just need to use the rule function again: The problem you can see here is that once Mathematica does the symbolic computation, it basically keeps everything there and perhaps more than that you are aware of. Moreover the dependances of the radicals and wrapped Log and fractional power really makes a direct way cheesy. 

Wow! You know way more than I do about Mathematica. I will have to study your approach.

One point is that my limits need to remain Sqrt functions, since I am performing a specific volume integral and the limits are a sphere as described in cylindrical coordinates. Consequently, I will have to investigate whether your approach as changed the integral into something that is not relevant to the scenario I am modeling.

Other than that, your attention to this issue is much appreciated. I will post back what I find...it will take a while...days? Weeks? But it is something to pursue where there was nothing before.

Thanks.

I have played around with your approach. It appears that while it works, there are other integrals in the model that cannot be solved symbolically. These occur in the last integrals over the "w" function. I may be able to do this numerically.

On the other hand, your rule and replacement approach will allow me to develop a power series expansion. The "d" in the expression is a function of z1 and z2. I can pull it out of the Sqrt function, which leaves me with 1/Sqrt[1+a] where for most values, a<1 and the series converges. However, I need to actually replace the "a" in the power series by the complete expression and allow Mathematica to do the algebra for me. I can integrate these term by term with simple integrals. It is tedious until I learn how to use other capabilities within Mathematica.

I have run across two other issues, however, with the power series expansion. First, there does not seem to be any one variable about which I can perform the expansion without using the symbol "a" to replace all functions at once.

The second thing is that the variables within the numerator allow the complete integrated series to converge term by term even when I chose values for "d" which would not allow the power series to actually converge. The total integral is more than the sum of its parts as far as the stability of the model is concerned. I did not expect this. What appears to be happening is that each term, in the limit, after being integrated comes to a finite value that is less than one.

Regards,