\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \! \left (\frac {2-5 \,{\mathrm e}^{\frac {3 x}{4}}}{4 \sqrt {-2+{\mathrm e}^{\frac {3 x}{4}}+{\mathrm e}^{\frac {3 x}{2}}}}\right )}{3} \]
command
integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4 \right |}\right ) + \frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} \right |}\right ) \]