\[ \int \frac {1}{8} \left (5 x+15 x^2+10 x^3+e^{2 x} \left (30 x+45 x^2\right )+e^x \left (-25 x-55 x^2-15 x^3\right )+\left (e^{2 x} \left (-25 x-30 x^2\right )+e^x \left (10 x+20 x^2+5 x^3\right )\right ) \log (2 x)+e^{2 x} \left (5 x+5 x^2\right ) \log ^2(2 x)\right ) \, dx \]
Optimal antiderivative \[ \frac {5 \left (x +1-{\mathrm e}^{x} \left (3-\ln \left (2 x \right )\right )\right )^{2} x^{2}}{16} \]
command
integrate(1/8*(5*x^2+5*x)*exp(x)^2*log(2*x)^2+1/8*((-30*x^2-25*x)*exp(x)^2+(5*x^3+20*x^2+10*x)*exp(x))*log(2*x)+1/8*(45*x^2+30*x)*exp(x)^2+1/8*(-15*x^3-55*x^2-25*x)*exp(x)+5/4*x^3+15/8*x^2+5/8*x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {5}{16} \, x^{2} e^{\left (2 \, x\right )} \log \left (2 \, x\right )^{2} + \frac {5}{16} \, x^{4} + \frac {5}{8} \, x^{3} - \frac {5}{8} \, x^{2} e^{x} - \frac {5}{32} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \log \left (2 \, x\right ) + \frac {5}{16} \, x^{2} + \frac {15}{32} \, {\left (6 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {15}{16} \, x e^{\left (2 \, x\right )} - \frac {5}{8} \, {\left (3 \, x^{3} + 2 \, x^{2} + x - 1\right )} e^{x} + \frac {5}{8} \, x e^{x} - \frac {5}{32} \, {\left ({\left (12 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} + x^{2}\right )} e^{x}\right )} \log \left (2 \, x\right ) - \frac {15}{32} \, e^{\left (2 \, x\right )} - \frac {5}{8} \, e^{x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {5}{8} \, {\left (x^{2} + x\right )} e^{\left (2 \, x\right )} \log \left (2 \, x\right )^{2} + \frac {5}{4} \, x^{3} + \frac {15}{8} \, x^{2} + \frac {15}{8} \, {\left (3 \, x^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} - \frac {5}{8} \, {\left (3 \, x^{3} + 11 \, x^{2} + 5 \, x\right )} e^{x} - \frac {5}{8} \, {\left ({\left (6 \, x^{2} + 5 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{3} + 4 \, x^{2} + 2 \, x\right )} e^{x}\right )} \log \left (2 \, x\right ) + \frac {5}{8} \, x\,{d x} \]________________________________________________________________________________________