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Please try the following:
int(int((x+y)^(1/2), x, 0, 1),y,0,1)

This fails with this message:
"Warning: unable to find taylorx63 integer solutions for taylorx63>=-1 and <=0, answer may be wrong. If you are computing an integral with exact boundaries, replace by approx. boundaries
[undef}
Unable to find singular points of antiderivative"

I tried to do replace the boundaries with approx values (0.0 and 1.0) as suggested in the warning, but this did not help.
This is a bug, I'm fixing it in Xcas and submitting the patch to HP. Until it is in, you can do that:
f1:=int((x+y)^(1/2), x);
int(subst(f1,x=1)-subst(f1,x=0),y,0,1);
The indefinite double integral works out OK on the Prime, so another workaround would be to calculate that and then use PREVAL twice (once for x, and once for y), which returns (16*sqrt(2)-8)/15.

Interestingly, the HP50G has no problem in exact mode with the definite double integral between 0 and 1, but hangs indefinitely in approximate mode.
Hi Parisse, this is good news. Thanks for that.
If I am right then you are the main developer of Xcas. May I ask how your relation to HP is? As far as I understood you are not employed by HP, but by University of Grenoble. However Xcas still seems to be deeply connected to HP calculators. The Xcas website does not go into much detail on that topic. It's just pure curiosity.
And since Xcas is used in the Prime and the License is GPLv3. What does this mean for the rest of the Prime's software? Does HP use a different license? I vaguely remember that GPLv3 demands that software which links against GPLv3 licensed software needs to be open sourced as well. I was thinking about whether this might have been a reason why the Prime's CAS is somewhat separated from the rest.

Thanks again
Giac/Xcas can be dual-licensed for commercial projects who do not want to release their code under the GPL license. This must obviously be the case for the HP Prime...
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