\[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{5}+1\right )^{\frac {2}{3}} \left (2 x^{5}+5 x^{3}+2\right )}{10 x^{5}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {1}{3}} \left (x^{5}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{2}+\frac {\ln \left (-x +2^{\frac {1}{3}} \left (x^{5}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{2}-\frac {\ln \left (x^{2}+2^{\frac {1}{3}} x \left (x^{5}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{5}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{4} \]
command
integrate((x^5+1)^(2/3)*(2*x^5-3)*(2*x^5+x^3+2)/x^6/(2*x^5-x^3+2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \, {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )} - 12 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{5} - x^{3} + 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{5} + 5 \, x^{3} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{120 \, x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]