Integral Hangs

[parisse]

What kind of third root did you take?
f:=integrate((x^2 +2*x)/(x^3 +3*x^2 +1)^(1/3),x)
returns ((x^3+3*x^2+1)^(1/3))^2/2.
But at x=-4, the argument of the ^(1/3) is negative and this returns a complex number.