\[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \]
Optimal antiderivative \[ -\frac {16 \left (-a e g +c d f \right )^{2} \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{35 c^{4} d^{4} e \sqrt {e x +d}}+\frac {12 \left (-a e g +c d f \right ) \left (g x +f \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{35 c^{2} d^{2} \sqrt {e x +d}}+\frac {2 \left (g x +f \right )^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{7 c d \sqrt {e x +d}}+\frac {16 g \left (-a e g +c d f \right )^{2} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{35 c^{3} d^{3} e} \]
command
integrate((g*x+f)^3*(e*x+d)^(1/2)/(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
\[ \frac {2 \, {\left (c^{3} d^{3} f^{3} - 3 \, a c^{2} d^{2} f^{2} g e + 3 \, a^{2} c d f g^{2} e^{2} - a^{3} g^{3} e^{3}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{4} d^{4}} + \frac {2 \, {\left (5 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g^{3} - 21 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} f g^{2} e + 35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} f^{2} g e^{2} + 6 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} g^{3} e^{2} - 35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3} f^{3} e^{3} - 28 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} f g^{2} e^{3} + 70 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} f^{2} g e^{4} + 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} g^{3} e^{4} - 56 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d f g^{2} e^{5} + 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} g^{3} e^{6}\right )} e^{\left (-4\right )}}{35 \, c^{4} d^{4}} + \frac {2 \, {\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} f^{2} g e^{4} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d f g^{2} e^{5} + 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d f g^{2} e^{2} + 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} g^{3} e^{6} - 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a g^{3} e^{3} + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} g^{3}\right )} e^{\left (-7\right )}}{35 \, c^{4} d^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {e x + d} {\left (g x + f\right )}^{3}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]________________________________________________________________________________________