\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d f -2 b c f +3 b d e \right ) \arctanh \left (\frac {\sqrt {b}\, \sqrt {d x +c}}{\sqrt {-a d +b c}}\right ) \sqrt {-a d +b c}}{b^{\frac {3}{2}} \left (-a f +b e \right )^{2}}+\frac {2 \left (-c f +d e \right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {f}\, \sqrt {d x +c}}{\sqrt {-c f +d e}}\right )}{\left (-a f +b e \right )^{2} \sqrt {f}}-\frac {\left (-a d +b c \right ) \sqrt {d x +c}}{b \left (-a f +b e \right ) \left (b x +a \right )} \]
command
integrate((d*x+c)**(3/2)/(b*x+a)**2/(f*x+e),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________