\[ \int \frac {(d+e x)^{3/2} (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (g x +f \right )^{2} \sqrt {e x +d}}{c d \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {8 g \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c^{3} d^{3} e \sqrt {e x +d}}+\frac {8 g^{2} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c^{2} d^{2} e} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)^2/(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
\[ -\frac {2 \, {\left (c^{2} d^{4} g^{2} - 6 \, c^{2} d^{3} f g e - 3 \, c^{2} d^{2} f^{2} e^{2} + 4 \, a c d^{2} g^{2} e^{2} + 12 \, a c d f g e^{3} - 8 \, a^{2} g^{2} e^{4}\right )} e^{\left (-1\right )}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3}} - \frac {2 \, {\left (c^{2} d^{2} f^{2} e - 2 \, a c d f g e^{2} + a^{2} g^{2} e^{3}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {2 \, {\left (6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{7} f g e^{8} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} g^{2} e^{9} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} g^{2} e^{6}\right )} e^{\left (-9\right )}}{3 \, c^{9} d^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________