\[ \int \frac {(d+e x)^{5/2} (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{2}}{3 c d \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {8 g \left (-a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {8 g \left (2 a \,e^{2} g -c d \left (d g +e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c^{3} d^{3} \left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}} \]
command
integrate((e*x+d)^(5/2)*(g*x+f)^2/(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
\[ \frac {2 \, {\left (3 \, c^{2} d^{4} g^{2} + 6 \, c^{2} d^{3} f g e - c^{2} d^{2} f^{2} e^{2} - 12 \, a c d^{2} g^{2} e^{2} - 4 \, a c d f g e^{3} + 8 \, a^{2} g^{2} e^{4}\right )}}{3 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{4} d^{5} - \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{3} e^{2}\right )}} + \frac {2 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g^{2} e^{\left (-1\right )}}{c^{3} d^{3}} - \frac {2 \, {\left (c^{2} d^{2} f^{2} e^{3} - 2 \, a c d f g e^{4} + 6 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d f g e + a^{2} g^{2} e^{5} - 6 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a g^{2} e^{2}\right )}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________