\[ \int \frac {625+9225 x+46950 x^2+78550 x^3+753 x^4+3798 x^5+6266 x^6+15 x^7+75 x^8+125 x^9+e^8 \left (x^4+15 x^5+75 x^6+125 x^7\right )+e^4 \left (-50 x^2-750 x^3-3783 x^4-6267 x^5-30 x^6-150 x^7-250 x^8\right )}{625+9375 x+46950 x^2+78550 x^3+750 x^4+3753 x^5+6266 x^6+15 x^7+75 x^8+125 x^9+e^8 \left (x^4+15 x^5+75 x^6+125 x^7\right )+e^4 \left (-50 x^2-750 x^3-3753 x^4-6267 x^5-30 x^6-150 x^7-250 x^8\right )} \, dx \]
Optimal antiderivative \[ x -\ln \left (\frac {3}{\left (x +\frac {25}{x^{2}}-{\mathrm e}^{4}\right ) \left (1+5 x \right )^{2}}+1\right ) \]
command
integrate(((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3783*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3798*x^5+753*x^4+78550*x^3+46950*x^2+9225*x+625)/((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3753*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3753*x^5+750*x^4+78550*x^3+46950*x^2+9375*x+625),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ x - \log \left ({\left | 25 \, x^{5} - 25 \, x^{4} e^{4} + 10 \, x^{4} - 10 \, x^{3} e^{4} + x^{3} - x^{2} e^{4} + 628 \, x^{2} + 250 \, x + 25 \right |}\right ) + \log \left ({\left | x^{3} - x^{2} e^{4} + 25 \right |}\right ) + 2 \, \log \left ({\left | 5 \, x + 1 \right |}\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {125 \, x^{9} + 75 \, x^{8} + 15 \, x^{7} + 6266 \, x^{6} + 3798 \, x^{5} + 753 \, x^{4} + 78550 \, x^{3} + 46950 \, x^{2} + {\left (125 \, x^{7} + 75 \, x^{6} + 15 \, x^{5} + x^{4}\right )} e^{8} - {\left (250 \, x^{8} + 150 \, x^{7} + 30 \, x^{6} + 6267 \, x^{5} + 3783 \, x^{4} + 750 \, x^{3} + 50 \, x^{2}\right )} e^{4} + 9225 \, x + 625}{125 \, x^{9} + 75 \, x^{8} + 15 \, x^{7} + 6266 \, x^{6} + 3753 \, x^{5} + 750 \, x^{4} + 78550 \, x^{3} + 46950 \, x^{2} + {\left (125 \, x^{7} + 75 \, x^{6} + 15 \, x^{5} + x^{4}\right )} e^{8} - {\left (250 \, x^{8} + 150 \, x^{7} + 30 \, x^{6} + 6267 \, x^{5} + 3753 \, x^{4} + 750 \, x^{3} + 50 \, x^{2}\right )} e^{4} + 9375 \, x + 625}\,{d x} \]________________________________________________________________________________________