i think i have done this

integrating by parts

I= root (1+x^3) int. 1dx - int. [d/dx(root(1+x^3).int. 1.dx]

I=x.root (1+x^3) - 3/2int. [x^3/root(1+x^3)]

take 1+x^3=t^2

3x^2dx=2t  dt

dx=2t/3x^2 dt

putting dx

I=x.root(1+x^3) - int. xdt

I=x.root(1+x^3) - int.(t^2-1)^1/3 dt

can anyone do after this.