How do we integrate $x\sqrt{x^3 + 1}$? Tried to use integration by parts but just keep going round in circles. Any help appreciated, thanks.

a possible answer can be found by Mathematica:
$$\frac{2 \left(x^5+x^2-3^{3/4} \sqrt{-\sqrt[6]{-1} \left(x+(-1)^{2/3}\right)}
   \sqrt{(-1)^{2/3} x^2+\sqrt[3]{-1} x+1} \left((-1)^{5/6} F\left(\sin
   ^{-1}\left(\frac{\sqrt{-(-1)^{5/6}
   (x+1)}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)+\sqrt{3} E\left(\sin
   ^{-1}\left(\frac{\sqrt{-(-1)^{5/6}
   (x+1)}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)\right)\right)}{7
   \sqrt{x^3+1}}$$
