\[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {4 e x +d}{\sqrt {3 d^{2}-2 \sqrt {-64 a \,e^{3}+d^{4}}}}\right )}{\sqrt {-64 a \,e^{3}+d^{4}}\, \sqrt {3 d^{2}-2 \sqrt {-64 a \,e^{3}+d^{4}}}}-\frac {2 \arctanh \left (\frac {4 e x +d}{\sqrt {3 d^{2}+2 \sqrt {-64 a \,e^{3}+d^{4}}}}\right )}{\sqrt {-64 a \,e^{3}+d^{4}}\, \sqrt {3 d^{2}+2 \sqrt {-64 a \,e^{3}+d^{4}}}} \]
command
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} + \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} - \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} + \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} - \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \]________________________________________________________________________________________