\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 g \left (e x +d \right )^{\frac {3}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{5 c d e}-\frac {4 \left (-a \,e^{2}+c \,d^{2}\right ) \left (4 a \,e^{2} g -c d \left (-d g +5 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{15 c^{3} d^{3} e \sqrt {e x +d}}-\frac {2 \left (4 a \,e^{2} g -c d \left (-d g +5 e f \right )\right ) \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{15 c^{2} d^{2} e} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)/(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
\[ \frac {2 \, {\left (c^{2} d^{3} f - a c d^{2} g e - a c d f e^{2} + a^{2} g e^{3}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{3} d^{3}} + \frac {4 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} g - 5 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{3} f e + 3 \, \sqrt {-c d^{2} e + a e^{3}} a c d^{2} g e^{2} + 5 \, \sqrt {-c d^{2} e + a e^{3}} a c d f e^{3} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} g e^{4}\right )} e^{\left (-2\right )}}{15 \, c^{3} d^{3}} + \frac {2 \, {\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} g e + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d f e^{2} - 10 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a g e^{3} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} g\right )} e^{\left (-4\right )}}{15 \, c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]________________________________________________________________________________________