Thank you both for your replies.  I'm not sure I follow the discussion to 
be honest as to how it applies to my original problem.  In particular: 

> Distilling this down you want to compute the integral of the square
>> root of a quadratic
>
> I'm not sure that is accurate.  If you are just referring to that it is 
(foo(xi))**2  + (bar(xi))**2, then yes the expression is quadratic in foo 
and bar.  But in general, foo(xi) and bar(xi) are themselves higher degree 
polynomials of xi (and in higher dimensions other coordinates too).  This 
is a very simple and minimal reproducing example: in this case foo and bar 
are linear polynomials so the whole expression is quadratic.  I have 
expressions I need to integrate, where foo(xi) and bar(xi) are higher-order 
polynomials terms of xi.

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