\[ \int \frac {-156-65 x+e^{2 x/3} (-36+12 x)+e^{x/3} \left (-150-6 x+10 x^2\right )+\left (25+e^{x/3} (12-2 x)+5 x\right ) \log (x)-\log ^2(x)}{2 x^3} \, dx \]
Optimal antiderivative \[ \frac {\left (12+6 \,{\mathrm e}^{\frac {x}{3}}+5 x -\ln \left (x \right )\right )^{2}}{4 x^{2}} \]
command
integrate(1/2*(-log(x)^2+((-2*x+12)*exp(1/3*x)+25+5*x)*log(x)+(12*x-36)*exp(1/3*x)^2+(10*x^2-6*x-150)*exp(1/3*x)-65*x-156)/x^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {60 \, x e^{\left (\frac {1}{3} \, x\right )} - 10 \, x \log \left (3\right ) - 12 \, e^{\left (\frac {1}{3} \, x\right )} \log \left (3\right ) + \log \left (3\right )^{2} - 10 \, x \log \left (\frac {1}{3} \, x\right ) - 12 \, e^{\left (\frac {1}{3} \, x\right )} \log \left (\frac {1}{3} \, x\right ) + 2 \, \log \left (3\right ) \log \left (\frac {1}{3} \, x\right ) + \log \left (\frac {1}{3} \, x\right )^{2} + 120 \, x + 36 \, e^{\left (\frac {2}{3} \, x\right )} + 144 \, e^{\left (\frac {1}{3} \, x\right )} - 24 \, \log \left (3\right ) - 24 \, \log \left (\frac {1}{3} \, x\right ) + 144}{4 \, x^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {12 \, {\left (x - 3\right )} e^{\left (\frac {2}{3} \, x\right )} + 2 \, {\left (5 \, x^{2} - 3 \, x - 75\right )} e^{\left (\frac {1}{3} \, x\right )} - {\left (2 \, {\left (x - 6\right )} e^{\left (\frac {1}{3} \, x\right )} - 5 \, x - 25\right )} \log \left (x\right ) - \log \left (x\right )^{2} - 65 \, x - 156}{2 \, x^{3}}\,{d x} \]________________________________________________________________________________________