Integral problems with roots of polynomials

[thereddevils]

Homework Statement

How do i solve this integral ?

$$\int \big( \sqrt{x^{3}+1} + \sqrt[3] {x^{2}+2x} \big) \ dx$$

Homework Equations

The Attempt at a Solution

what is the appropriate substitution to make here

 

[irycio]

There is no antiderivative of this function that could be written down using elementary functions only. It takes some hypergeometric and elliptic functions, type it down into wolfram alpha and see what you get.