Exercise
$\int_{x^2}^0\sqrt[2]{1+y^3}dy$
Step-by-step Solution
1
The integral $\int\sqrt{1+y^3}dy$ is non-elementary
$\left[\frac{2}{5\sqrt{y^3+1}}\left(y^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(y+1\right)}\sqrt{y^2-y+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(y+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+y\right)\right]_{x^2}^{0}$
Final answer to the exercise
$\left[\frac{2}{5\sqrt{y^3+1}}\left(y^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(y+1\right)}\sqrt{y^2-y+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(y+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+y\right)\right]_{x^2}^{0}$