\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 c \left (-4 b e g +3 c d g +5 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{4 e^{2} \left (-b e +2 c d \right )^{\frac {7}{2}}}+\frac {d g -e f}{2 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {4 b e g -3 c d g -5 c e f}{4 e^{2} \left (-b e +2 c d \right )^{2} \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {3 c \left (-4 b e g +3 c d g +5 c e f \right ) \sqrt {e x +d}}{4 e^{2} \left (-b e +2 c d \right )^{3} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^(3/2)/(-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
\[ \frac {3 \, {\left (3 \, c^{2} d g + 5 \, c^{2} f e - 4 \, b c g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, {\left (c^{2} d g + c^{2} f e - b c g e\right )}}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} + \frac {2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 18 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e + 7 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 9 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2} - 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c g e^{2} + 7 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} f e - 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c g e}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left (x e + d\right )}^{2} c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________