\[ \int \frac {(d+e x)^{3/2} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \]
Optimal antiderivative \[ \frac {8 \left (-a e g +c d f \right ) \left (6 a \,e^{2} g +c d \left (-7 d g +e f \right )\right ) \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{105 c^{4} d^{4} e g \sqrt {e x +d}}-\frac {2 \left (6 a \,e^{2} g +c d \left (-7 d g +e f \right )\right ) \left (g x +f \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{35 c^{2} d^{2} g \sqrt {e x +d}}+\frac {2 e \left (g x +f \right )^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{7 c d g \sqrt {e x +d}}-\frac {8 \left (-a e g +c d f \right ) \left (6 a \,e^{2} g +c d \left (-7 d g +e f \right )\right ) \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{105 c^{3} d^{3} e} \]
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)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (c^{3} d^{4} f^{2} - 2 \, a c^{2} d^{3} f g e - a c^{2} d^{2} f^{2} e^{2} + a^{2} c d^{2} g^{2} e^{2} + 2 \, a^{2} c d f g e^{3} - a^{3} g^{2} e^{4}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{4} d^{4}} - \frac {4 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g^{2} - 14 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} f g e + 35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} f^{2} e^{2} + 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} g^{2} e^{2} - 42 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} f g e^{3} - 35 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} f^{2} e^{4} + 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} g^{2} e^{4} + 56 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d f g e^{5} - 24 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} g^{2} e^{6}\right )} e^{\left (-3\right )}}{105 \, c^{4} d^{4}} + \frac {2 \, {\left (70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{3} f g e^{3} + 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} f^{2} e^{4} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d^{2} g^{2} e^{4} + 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d^{2} g^{2} e - 140 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d f g e^{5} + 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d f g e^{2} + 105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} g^{2} e^{6} - 63 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a g^{2} e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} g^{2}\right )} e^{\left (-6\right )}}{105 \, c^{4} d^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]________________________________________________________________________________________