Something is wrong:
(%i11) integrate(x^(0.5) * (1+x^2)^(0.5),x,0,1);
(%o11) (%e^(3/4)*sqrt(%pi)*%i*erf(%i/2))/2-(%e^(3/4)*sqrt(%pi)*%i*erf((3*%i)/2)-2^(3/2))/2
(%i12) float(%);
(%o12) 8.862088383369269
Numerical check:
(%i15) quad_qag (x^(0.5) * (1+x^2)^(0.5), x, 0, 1, 3, 'epsrel=5d-8);
(%o15) [0.79248561602668,2.6069049931541242*10^-8,651,0]
Nounform without floats:
(%i16) integrate(x^(1/2) * (1+x^2)^(1/2),x,0,1);
(%o16) integrate(sqrt(x)*sqrt(x^2+1),x,0,1)