\[ \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{3}}{3 c d \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {4 g \left (g x +f \right )^{2} \sqrt {e x +d}}{c^{2} d^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {16 g^{2} \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c^{4} d^{4} e \sqrt {e x +d}}+\frac {16 g^{3} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c^{3} d^{3} e} \]
command
integrate((e*x+d)^(5/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (c^{3} d^{6} g^{3} - 9 \, c^{3} d^{5} f g^{2} e - 9 \, c^{3} d^{4} f^{2} g e^{2} + 6 \, a c^{2} d^{4} g^{3} e^{2} + c^{3} d^{3} f^{3} e^{3} + 36 \, a c^{2} d^{3} f g^{2} e^{3} + 6 \, a c^{2} d^{2} f^{2} g e^{4} - 24 \, a^{2} c d^{2} g^{3} e^{4} - 24 \, a^{2} c d f g^{2} e^{5} + 16 \, a^{3} g^{3} e^{6}\right )}}{3 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{5} d^{6} e - \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{4} e^{3}\right )}} - \frac {2 \, {\left (c^{3} d^{3} f^{3} e^{3} - 3 \, a c^{2} d^{2} f^{2} g e^{4} + 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} f^{2} g e + 3 \, a^{2} c d f g^{2} e^{5} - 18 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c d f g^{2} e^{2} - a^{3} g^{3} e^{6} + 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} g^{3} e^{3}\right )}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4}} + \frac {2 \, {\left (9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{9} d^{9} f g^{2} e^{8} - 9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{8} d^{8} g^{3} e^{9} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{8} d^{8} g^{3} e^{6}\right )} e^{\left (-9\right )}}{3 \, c^{12} d^{12}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________