\[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{e^{2} \left (-b e +2 c d \right )^{\frac {3}{2}}}+\frac {2 \left (-b e g +c d g +c e f \right ) \sqrt {e x +d}}{c \,e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
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)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (d g - f e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d e^{2} - b e^{3}\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, {\left (\sqrt {2 \, c d - b e} c d g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - \sqrt {2 \, c d - b e} c f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e - \sqrt {-2 \, c d + b e} c d g - \sqrt {-2 \, c d + b e} c f e + \sqrt {-2 \, c d + b e} b g e\right )}}{2 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{2} d e^{2} - \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c e^{3}} + \frac {2 \, {\left (c d g + c f e - b g e\right )}}{{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________