\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{2}}-\frac {\left (b e g -4 c d g +2 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{2 e^{2} \sqrt {c}}-\frac {\left (b e g -4 c d g +2 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \left (-b e +2 c d \right )} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -{\left (\frac {{\left (4 \, c d g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, c f e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b g e \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}}}{\sqrt {c}}\right ) e^{\left (-3\right )}}{\sqrt {c}} - 2 \, {\left (\sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} f e \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-3\right )} - \frac {{\left (2 \, c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} g e \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-3\right )}}{\frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________