Exercise
$\int_{-1}^x\sqrt{t^3+1}dt$
Step-by-step Solution
1
The integral $\int\sqrt{t^3+1}dt$ is non-elementary
$\left[\frac{2}{5\sqrt{t^3+1}}\left(t^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(t+1\right)}\sqrt{t^2-t+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(t+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+t\right)\right]_{-1}^{x}$
Final answer to the exercise
$\left[\frac{2}{5\sqrt{t^3+1}}\left(t^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(t+1\right)}\sqrt{t^2-t+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(t+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+t\right)\right]_{-1}^{x}$