Integral dx/sqrt(1+x^3)

[Helge Gabert]

While the Prime cannot symbolically integrate the indefinite integral

1/sqrt(1+x^3) dx

which gets understandably complicated (involving an elliptic integral of the first kind),

in CAS, when trying to compute the definite integral between 0 and infinity, which exists and equals 2*Gamma(1/3)*Gamma(7/6)/sqrt(pi),

the Prime returns via approx(int(1/sqrt(1+x^3),x,0,inf)) [3184.6... 2.804 ...], where 2.804 is very close to the solution. The message is "adaptive method failure, will try with Romberg."

So, int (Gaussian quadrature?) seems to work, even though an error message (adaptive method failure) is issued?

Any insights are appreciated.

P.S. Using romberg instead of int returns a bracket which is totally off [6366... 3184...], but here the CAS message "unable to find numeric integral" makes sense.