Ejercicio
$\int_1^3\left(\sqrt{x^3+1}\right)dx$
Solución explicada paso por paso
1
La integral $\int\sqrt{x^3+1}dx$ es no elemental
$\left[\frac{2}{5\sqrt{x^3+1}}\left(x^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(x+1\right)}\sqrt{x^2-x+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(x+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+x\right)\right]_{1}^{3}$
Respuesta final al problema
$\left[\frac{2}{5\sqrt{x^3+1}}\left(x^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(x+1\right)}\sqrt{x^2-x+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(x+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+x\right)\right]_{1}^{3}$