\[ \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-2 b e g +c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 c^{2} e^{2} \sqrt {e x +d}}-\frac {2 g \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 c \,e^{2}} \]
command
integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{3} \, {\left (\frac {{\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} g e^{\left (-1\right )}}{c^{2}} - \frac {3 \, {\left (c d g + c f e - b g e\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} e^{\left (-1\right )}}{c^{2}} + \frac {{\left (\sqrt {2 \, c d - b e} c d g + 3 \, \sqrt {2 \, c d - b e} c f e - 2 \, \sqrt {2 \, c d - b e} b g e\right )} e^{\left (-1\right )}}{c^{2}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________