\[ \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctan \left (\frac {\sqrt {g}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a e g +c d f}\, \sqrt {e x +d}}\right ) \sqrt {g}}{\left (-a e g +c d f \right )^{\frac {3}{2}}}-\frac {2 \sqrt {e x +d}}{\left (-a e g +c d f \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate((e*x+d)^(3/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -2 \, {\left (\frac {g \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e} {\left (c d f - a g e\right )}} + \frac {1}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} {\left (c d f - a g e\right )}}\right )} e + \frac {2 \, {\left (\sqrt {-c d^{2} e + a e^{3}} g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) + \sqrt {c d f g - a g^{2} e} e\right )}}{\sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} c d f - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a g e} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________