\[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx \]
Optimal antiderivative \[ x^{2} \left (4 x -{\mathrm e}+\left (\frac {2 x}{-x +{\mathrm e}^{3}}+\frac {x}{5}+\frac {1}{5}\right )^{2}\right ) \]
command
integrate(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*exp(3)-25*x^3),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{25} \, x^{4} + \frac {82}{25} \, x^{3} - \frac {4}{5} \, x^{2} e^{3} - x^{2} e + \frac {81}{25} \, x^{2} - \frac {4}{5} \, x e^{6} + \frac {36}{5} \, x e^{3} - \frac {4 \, {\left (x e^{12} - 19 \, x e^{9} - e^{15} + 14 \, e^{12}\right )}}{5 \, {\left (x - e^{3}\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {2 \, {\left (2 \, x^{6} + 123 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} - {\left (2 \, x^{3} + 153 \, x^{2} - 25 \, x e + x\right )} e^{9} + {\left (6 \, x^{4} + 419 \, x^{3} - 75 \, x^{2} e - 27 \, x^{2}\right )} e^{6} - {\left (6 \, x^{5} + 389 \, x^{4} - 75 \, x^{3} e + 153 \, x^{3}\right )} e^{3}\right )}}{25 \, {\left (x^{3} - 3 \, x^{2} e^{3} + 3 \, x e^{6} - e^{9}\right )}}\,{d x} \]________________________________________________________________________________________