integrate(sqrt(x^2+y^2),x,0,1)$ integrate(%,y,0,1)$



(%i1) algebraic:true$

 Let S = 'integrate(integrate(sqrt(x^2+y^2),x,0,1),y,0,1).
After change variables  x=r*cos(phi), y=r*sin(phi) we get

(%i2) integrate(integrate(r^2,r,0,1/cos(phi)),phi, 0, %pi/4)+
        integrate(integrate(r^2,r,0,1/sin(phi)),phi,%pi/4,%pi/2)$

(%i3) radcan(%);
(%o3) (log(sqrt(2)+2)-log(2-sqrt(2))+2^(3/2))/6
(%i4) logcontract(%);
(%o4) (log(-(sqrt(2)+2)/(sqrt(2)-2))+2^(3/2))/6

(%i5) S:ratsimp(%);
(%o5) (log(2^(3/2)+3)+2^(3/2))/6

(%i6) float(S), numer;
(%o6) 0.76519571646421

(%i7) S1:1/3*(sqrt(2)+log(sqrt(2)+1));
(%o7) (log(sqrt(2)+1)+sqrt(2))/3
(%i8) float(S1), numer;
(%o8) 0.76519571646421

(%i9) S-S1;
(%o9) (log(2^(3/2)+3)+2^(3/2))/6-(log(sqrt(2)+1)+sqrt(2))/3
(%i10) logcontract(%),ratsimp;
(%o10) 0
 Then  S=S1

best

Aleksas D