(1)

\begin{align*} \int\frac{x^{\alpha}}{x^{\beta}+\gamma}dx & =\frac{1}{\gamma}\int\frac{x^{\alpha}}{1+\frac{x^{\beta}}{\gamma}}dx\\ & =\frac{1}{\gamma}\int x^{\alpha}F\left(1;;-\frac{x^{\beta}}{\gamma}\right)dx\\ & =\frac{x^{\alpha+1}}{\left(\alpha+1\right)\gamma}F\left(1,\frac{\alpha+1}{\beta};\frac{\alpha+1}{\beta}+1;-\frac{x^{\beta}}{\gamma}\right)+C \end{align*}

(2)

\[ \begin{align*}\int_{0}^{\infty}\frac{x^{\alpha}}{x^{n}+1}dx & =\frac{1}{n}\int_{0}^{\infty}\frac{y^{\frac{\alpha}{n}+\frac{1}{n}-1}}{y+1}dy\cmt{y=x^{n}}\\ & =\frac{1}{n}\int_{1}^{0}z\left(\frac{1-z}{z}\right)^{\frac{\alpha+1}{n}-1}\left(-\frac{1}{z^{2}}\right)dz\cmt{z=\frac{1}{y+1},y=\frac{1-z}{z},dy=-\frac{1}{z^{2}}dz}\\ & =\frac{1}{n}\int_{0}^{1}z^{-\frac{\alpha+1}{n}}\left(1-z\right)^{\frac{\alpha+1}{n}-1}dz\\ & =\frac{1}{n}B\left(1-\frac{\alpha+1}{n},\frac{\alpha+1}{n}\right)\\ & =\frac{1}{n}\Gamma\left(1-\frac{\alpha+1}{n}\right)\Gamma\left(\frac{\alpha+1}{n}\right)\\ & =\frac{\pi}{n}\sin^{-1}\left(\frac{\left(\alpha+1\right)\pi}{n}\right) \end{align*} \]