\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-2 b e g +c d g +3 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{e^{2} \left (-b e +2 c d \right )^{\frac {5}{2}}}+\frac {d g -e f}{e^{2} \left (-b e +2 c d \right ) \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {\left (-2 b e g +c d g +3 c e f \right ) \sqrt {e x +d}}{e^{2} \left (-b e +2 c d \right )^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^(1/2)/(-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
\[ \frac {{\left (c d g + 3 \, c f e - 2 \, b g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (4 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} \sqrt {-2 \, c d + b e}} + \frac {4 \, c^{2} d^{2} g + 4 \, c^{2} d f e - 6 \, b c d g e + {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c d g - 2 \, b c f e^{2} + 2 \, b^{2} g e^{2} + 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c f e - 2 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b g e}{{\left (4 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} {\left (2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c d - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b e - {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________