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