Integral CAS approximation

[parisse]

HOME is using CAS for integral evaluation, but without trying an exact computation first (in other words it will call integrate(sqrt(1+x^3),x,-1.0,1.0)).
If you run integrate(sqrt(1+x^3),x,-1,1) in CAS, you get the initial integral expressed with other integrals because the CAS tried to get an exact answer, and this explains the small numeric differences, the general rule is that the precision is better without initial symbolic step (if the integral can not be expressed in closed form).