\[ \int \frac {-625-625 x+250 e^2 x-25 e^4 x+\left (1000 x-200 e^2 x\right ) \log (19)-400 x \log ^2(19)}{-15625+46875 x-46875 x^2+15625 x^3-30 e^{10} x^3+e^{12} x^3+e^2 \left (-18750 x+37500 x^2-18750 x^3\right )+e^6 \left (1500 x^2-2500 x^3\right )+e^8 \left (-75 x^2+375 x^3\right )+e^4 \left (1875 x-11250 x^2+9375 x^3\right )+\left (-75000 x+150000 x^2-75000 x^3-600 e^8 x^3+24 e^{10} x^3+e^4 \left (18000 x^2-30000 x^3\right )+e^6 \left (-1200 x^2+6000 x^3\right )+e^2 \left (15000 x-90000 x^2+75000 x^3\right )\right ) \log (19)+\left (30000 x-180000 x^2+150000 x^3-4800 e^6 x^3+240 e^8 x^3+e^2 \left (72000 x^2-120000 x^3\right )+e^4 \left (-7200 x^2+36000 x^3\right )\right ) \log ^2(19)+\left (96000 x^2-160000 x^3-19200 e^4 x^3+1280 e^6 x^3+e^2 \left (-19200 x^2+96000 x^3\right )\right ) \log ^3(19)+\left (-19200 x^2+96000 x^3-38400 e^2 x^3+3840 e^4 x^3\right ) \log ^4(19)+\left (-30720 x^3+6144 e^2 x^3\right ) \log ^5(19)+4096 x^3 \log ^6(19)} \, dx \]
Optimal antiderivative \[ \frac {x}{\left (5-\frac {x \left ({\mathrm e}^{2}-5+4 \ln \left (19\right )\right )^{2}}{5}\right )^{2}} \]
command
integrate((-400*x*log(19)^2+(-200*exp(2)*x+1000*x)*log(19)-25*x*exp(2)^2+250*exp(2)*x-625*x-625)/(4096*x^3*log(19)^6+(6144*x^3*exp(2)-30720*x^3)*log(19)^5+(3840*x^3*exp(2)^2-38400*x^3*exp(2)+96000*x^3-19200*x^2)*log(19)^4+(1280*x^3*exp(2)^3-19200*x^3*exp(2)^2+(96000*x^3-19200*x^2)*exp(2)-160000*x^3+96000*x^2)*log(19)^3+(240*x^3*exp(2)^4-4800*x^3*exp(2)^3+(36000*x^3-7200*x^2)*exp(2)^2+(-120000*x^3+72000*x^2)*exp(2)+150000*x^3-180000*x^2+30000*x)*log(19)^2+(24*x^3*exp(2)^5-600*x^3*exp(2)^4+(6000*x^3-1200*x^2)*exp(2)^3+(-30000*x^3+18000*x^2)*exp(2)^2+(75000*x^3-90000*x^2+15000*x)*exp(2)-75000*x^3+150000*x^2-75000*x)*log(19)+x^3*exp(2)^6-30*x^3*exp(2)^5+(375*x^3-75*x^2)*exp(2)^4+(-2500*x^3+1500*x^2)*exp(2)^3+(9375*x^3-11250*x^2+1875*x)*exp(2)^2+(-18750*x^3+37500*x^2-18750*x)*exp(2)+15625*x^3-46875*x^2+46875*x-15625),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {25 \, x}{{\left (8 \, x e^{2} \log \left (19\right ) + 16 \, x \log \left (19\right )^{2} + x e^{4} - 10 \, x e^{2} - 40 \, x \log \left (19\right ) + 25 \, x - 25\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________