\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{45 x^{9}}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {2 x^{8}-2 x^{4}-1}}\right ) 2^{\frac {3}{4}}}{4}-\frac {\arctanh \left (\frac {2 \,2^{\frac {1}{4}} x \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}}{2 x^{2}+\sqrt {2}\, \sqrt {2 x^{8}-2 x^{4}-1}}\right ) 2^{\frac {3}{4}}}{4} \]
command
integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {180 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \arctan \left (-\frac {32 \, x^{16} - 32 \, x^{8} + 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{9} - 8 \, x^{5} - x\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} + 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{11} - 8 \, x^{7} - 3 \, x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{10} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x^{5} + 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{16} - 40 \, x^{12} + 28 \, x^{8} + 20 \, x^{4} + 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{10} - 8 \, x^{6} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} + 32 \, {\left (2 \, x^{11} - x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + \sqrt {2} {\left (2 \, x^{8} - 1\right )}}{2 \, x^{8} - 1}} + 8}{8 \, {\left (4 \, x^{16} - 64 \, x^{12} + 60 \, x^{8} + 32 \, x^{4} + 1\right )}}\right ) - 180 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \arctan \left (-\frac {32 \, x^{16} - 32 \, x^{8} - 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{9} - 8 \, x^{5} - x\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} - 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{11} - 8 \, x^{7} - 3 \, x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{10} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x^{5} - 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{16} - 40 \, x^{12} + 28 \, x^{8} + 20 \, x^{4} + 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{10} - 8 \, x^{6} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} + 32 \, {\left (2 \, x^{11} - x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - \sqrt {2} {\left (2 \, x^{8} - 1\right )}}{2 \, x^{8} - 1}} + 8}{8 \, {\left (4 \, x^{16} - 64 \, x^{12} + 60 \, x^{8} + 32 \, x^{4} + 1\right )}}\right ) + 45 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + \sqrt {2} {\left (2 \, x^{8} - 1\right )}\right )}}{2 \, x^{8} - 1}\right ) - 45 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (-\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - \sqrt {2} {\left (2 \, x^{8} - 1\right )}\right )}}{2 \, x^{8} - 1}\right ) - 64 \, {\left (20 \, x^{16} - 4 \, x^{12} + 9 \, x^{8} + 2 \, x^{4} + 5\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{2880 \, x^{9}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]