\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-d g +e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right ) \sqrt {-b e +2 c d}}{e^{2}}+\frac {2 \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \sqrt {e x +d}} \]
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)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{3} \, {\left (\frac {3 \, {\left (2 \, c d^{2} g - 2 \, c d f e - b d g e + b f e^{2}\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d g - 3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} f e + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} g}{c^{3}}\right )} e^{\left (-2\right )} + \frac {2 \, {\left (6 \, c^{2} d^{2} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 6 \, c^{2} d f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e - 3 \, b c d g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e + 3 \, b c f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e^{2} + 5 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c d g - 3 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c f e - \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b g e\right )} e^{\left (-2\right )}}{3 \, \sqrt {-2 \, c d + b e} c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________