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