How to integrate $\frac{2+5x^3}{2x^3+2}\sqrt{x^3+1}$ How would one find the integral of  $\frac{2+5x^3}{2x^3+2}\sqrt{x^3+1}$ with respect to $x$?
I already know the antiderivative (from Wolfram Alpha), but I don't know how to integrate this function with pencil and paper.
Is there maybe a clever variable substitution?

HINT:
I would say that
$\frac{2+5x^3}{2x^3+2}\sqrt{x^3+1}=\frac{2+5x^3}{2(x^3+1)}\frac{x^3+1}{\sqrt{x^3+1}}=\frac{2+5x^3}{2\sqrt{x^3+1}}=\frac{2(x^3+1)+3x^3}{2\sqrt{x^3+1}}=\frac{x^3+1}{\sqrt{x^3+1}}+\frac{3x^3}{2\sqrt{x^3+1}}=$
$=\sqrt{x^3+1}+\frac{3x^3}{2\sqrt{x^3+1}}=1\cdot\sqrt{x^3+1}+x\cdot\frac{3x^2}{2\sqrt{x^3+1}}=\frac{d}{dx}(x\cdot \sqrt{x^3+1})$
