\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (x^{4}+2 x^{2}+1\right ) \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}+\frac {3 \,2^{\frac {1}{4}} \arctan \! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{4}-\frac {3 \arctan \! \left (\frac {2^{\frac {3}{4}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{\sqrt {2}\, x^{2}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {3}{4}}}{8}-\frac {3 \,2^{\frac {1}{4}} \operatorname {arctanh}\! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{4}-\frac {3 \,\operatorname {arctanh}\! \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{8} \]
command
Int[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {4 \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (-\frac {9}{8},1,\frac {3}{4},-\frac {1}{8},x^2,-x^2\right )}{3 \sqrt [4]{x^2+1} x^3}+\frac {4 \sqrt [4]{x^5+x^3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {15}{8},\frac {23}{8},-x^2\right )}{15 \sqrt [4]{x^2+1}}+\frac {4 \sqrt [4]{x^5+x^3} x \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{8},\frac {15}{8},-x^2\right )}{7 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {3}{4},\frac {7}{8},-x^2\right )}{\sqrt [4]{x^2+1} x}-\frac {8 \sqrt [4]{x^5+x^3} \operatorname {Hypergeometric2F1}\left (-\frac {9}{8},\frac {3}{4},-\frac {1}{8},-x^2\right )}{9 \sqrt [4]{x^2+1} x^3} \]