\[ \int \frac {x \left (2 e-2 f x^3\right )}{e^2+4 e f x^3+4 d f x^4+4 f^2 x^6} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {2 x^{2} \sqrt {d}\, \sqrt {f}}{2 f \,x^{3}+e}\right )}{2 \sqrt {d}\, \sqrt {f}} \]
command
integrate(x*(-2*f*x^3+2*e)/(4*f^2*x^6+4*d*f*x^4+4*e*f*x^3+e^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {-d f} \log \left ({\left | 2 \, f x^{3} + 2 \, \sqrt {-d f} x^{2} + e \right |}\right )}{4 \, d f} + \frac {\sqrt {-d f} \log \left ({\left | 2 \, f x^{3} - 2 \, \sqrt {-d f} x^{2} + e \right |}\right )}{4 \, d f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {2 \, {\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} + 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \]________________________________________________________________________________________