4402.sagews

# Here is the change of variables and the Jacobian
u,v = var('u,v')
Phi = Matrix([[(u/v)^(1/3)],[(v*u^2)^(1/3)]])
jacobian(Phi,[u,v])

[ 1/3*(1/v)/(u/v)^(2/3) -1/3*u/(v^2*(u/v)^(2/3))] [ 2/3*u*v/(u^2*v)^(2/3) 1/3*u^2/(u^2*v)^(2/3)]

# Here is the determinant
J = jacobian(Phi,[u,v])
det(J).simplify()

1/3*u^2/((u^2)^(2/3)*v^(5/3)*(u/v)^(2/3))

# Here is the integral
integral(integral(det(J),u,1,2),v,1,2)

1/3*log(2)