\[ \int \frac {36+48 x+16 x^2+4 e^3 x^2+e^{3/2} \left (24 x+16 x^2\right )+\left (2 x^2+e^{3/2} x^2\right ) \log (3)}{9 x^2+12 x^3+4 x^4+e^3 x^4+e^{3/2} \left (6 x^3+4 x^4\right )} \, dx \]
Optimal antiderivative \[ -2-\frac {4}{x}-\frac {\ln \left (3\right )}{3+x \,{\mathrm e}^{\frac {3}{2}}+2 x} \]
command
integrate(((x^2*exp(3/2)+2*x^2)*log(3)+4*x^2*exp(3/2)^2+(16*x^2+24*x)*exp(3/2)+16*x^2+48*x+36)/(x^4*exp(3/2)^2+(4*x^4+6*x^3)*exp(3/2)+4*x^4+12*x^3+9*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {4 \, x e^{\frac {3}{2}} + x \log \left (3\right ) + 8 \, x + 12}{x^{2} e^{\frac {3}{2}} + 2 \, x^{2} + 3 \, x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________