Integral de $$$\frac{x^{2}}{1 - x^{\sqrt{2}}}$$$

Encontre $$$\int \frac{x^{2}}{1 - x^{\sqrt{2}}}\, dx$$$.

Esta integral não possui forma fechada:

$${\color{red}{\int{\frac{x^{2}}{1 - x^{\sqrt{2}}} d x}}} = {\color{red}{\left(\frac{x^{3} {{}_{2}F_{1}\left(\begin{matrix} 1, \frac{3 \sqrt{2}}{2} \\ 1 + \frac{3 \sqrt{2}}{2} \end{matrix}\middle| {x^{\sqrt{2}}} \right)}}{3}\right)}}$$

Portanto,

$$\int{\frac{x^{2}}{1 - x^{\sqrt{2}}} d x} = \frac{x^{3} {{}_{2}F_{1}\left(\begin{matrix} 1, \frac{3 \sqrt{2}}{2} \\ 1 + \frac{3 \sqrt{2}}{2} \end{matrix}\middle| {x^{\sqrt{2}}} \right)}}{3}$$

Simplifique:

$$\int{\frac{x^{2}}{1 - x^{\sqrt{2}}} d x} = \frac{\sqrt{2} x^{3} \Phi\left(x^{\sqrt{2}}, 1, \frac{3 \sqrt{2}}{2}\right)}{2}$$

Adicione a constante de integração:

$$\int{\frac{x^{2}}{1 - x^{\sqrt{2}}} d x} = \frac{\sqrt{2} x^{3} \Phi\left(x^{\sqrt{2}}, 1, \frac{3 \sqrt{2}}{2}\right)}{2}+C$$

$$$\int \frac{x^{2}}{1 - x^{\sqrt{2}}}\, dx = \frac{\sqrt{2} x^{3} \Phi\left(x^{\sqrt{2}}, 1, \frac{3 \sqrt{2}}{2}\right)}{2} + C$$$A