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