\[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (\frac {d}{4 e}+x \right ) \left (13 d^{4}-256 a \,e^{3}-48 d^{2} e^{2} \left (\frac {d}{4 e}+x \right )^{2}\right )}{\left (-16384 a^{2} e^{6}-64 a \,d^{4} e^{3}+5 d^{8}\right ) \left (8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}\right )}-\frac {24 e \arctanh \left (\frac {4 e x +d}{\sqrt {3 d^{2}-2 \sqrt {-64 a \,e^{3}+d^{4}}}}\right ) \left (d^{4}+128 a \,e^{3}-d^{2} \sqrt {-64 a \,e^{3}+d^{4}}\right )}{\left (-64 a \,e^{3}+d^{4}\right )^{\frac {3}{2}} \left (256 a \,e^{3}+5 d^{4}\right ) \sqrt {3 d^{2}-2 \sqrt {-64 a \,e^{3}+d^{4}}}}+\frac {24 e \arctanh \left (\frac {4 e x +d}{\sqrt {3 d^{2}+2 \sqrt {-64 a \,e^{3}+d^{4}}}}\right ) \left (d^{4}+128 a \,e^{3}+d^{2} \sqrt {-64 a \,e^{3}+d^{4}}\right )}{\left (-64 a \,e^{3}+d^{4}\right )^{\frac {3}{2}} \left (256 a \,e^{3}+5 d^{4}\right ) \sqrt {3 d^{2}+2 \sqrt {-64 a \,e^{3}+d^{4}}}} \]
command
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________