\[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{2}}-\frac {2 \left (-3 b e g +4 c d g +2 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )} \]
command
integrate((g*x+f)/(e*x+d)^2/(-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
\[ \frac {2}{3} \, {\left (\frac {{\left (4 \, \sqrt {-c} c d g + 2 \, \sqrt {-c} c f e - 3 \, b \sqrt {-c} g e\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}} + \frac {\frac {{\left (3 \, c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}}\right )} d g e}{2 \, c d e - b e^{2}} - \frac {{\left (3 \, c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}}\right )} f e^{2}}{2 \, c d e - b e^{2}} - 3 \, \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} g}{2 \, c d e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________