\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \,2^{\frac {1}{4}} \arctan \! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}-\frac {\sqrt {2}\, \arctan \! \left (\frac {\sqrt {2}\, x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{5}+x^{3}}}\right )}{3}-\frac {2 \,2^{\frac {1}{4}} \operatorname {arctanh}\! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\! \left (\frac {\frac {\sqrt {2}\, x^{2}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, \sqrt {2}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3} \]
command
Int[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {8 x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},1,-\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{21 \sqrt [4]{x^2+1}} \]