\[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (g x +f \right )^{3} \sqrt {e x +d}}{c d \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {16 g \left (-a e g +c d f \right ) \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{5 c^{4} d^{4} e \sqrt {e x +d}}+\frac {12 g \left (g x +f \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{5 c^{2} d^{2} \sqrt {e x +d}}+\frac {16 g^{2} \left (-a e g +c d f \right ) \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{5 c^{3} d^{3} e} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)^3/(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
\[ \frac {2 \, {\left (c^{3} d^{6} g^{3} - 5 \, c^{3} d^{5} f g^{2} e + 15 \, c^{3} d^{4} f^{2} g e^{2} + 2 \, a c^{2} d^{4} g^{3} e^{2} + 5 \, c^{3} d^{3} f^{3} e^{3} - 20 \, a c^{2} d^{3} f g^{2} e^{3} - 30 \, a c^{2} d^{2} f^{2} g e^{4} + 8 \, a^{2} c d^{2} g^{3} e^{4} + 40 \, a^{2} c d f g^{2} e^{5} - 16 \, a^{3} g^{3} e^{6}\right )} e^{\left (-2\right )}}{5 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{4}} - \frac {2 \, {\left (c^{3} d^{3} f^{3} e - 3 \, a c^{2} d^{2} f^{2} g e^{2} + 3 \, a^{2} c d f g^{2} e^{3} - a^{3} g^{3} e^{4}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{4}} + \frac {2 \, {\left (15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{18} d^{18} f^{2} g e^{24} - 30 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{17} d^{17} f g^{2} e^{25} + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{17} d^{17} f g^{2} e^{22} + 15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{16} d^{16} g^{3} e^{26} - 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{16} d^{16} g^{3} e^{23} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{16} d^{16} g^{3} e^{20}\right )} e^{\left (-25\right )}}{5 \, c^{20} d^{20}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________