\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 c \left (-4 b e g +c d g +7 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{4 e^{2} \left (-b e +2 c d \right )^{\frac {9}{2}}}+\frac {d g -e f}{2 e^{2} \left (-b e +2 c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}+\frac {\left (-4 b e g +c d g +7 c e f \right ) \sqrt {e x +d}}{6 e^{2} \left (-b e +2 c d \right )^{2} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}-\frac {5 \left (-4 b e g +c d g +7 c e f \right )}{12 e^{2} \left (-b e +2 c d \right )^{3} \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {5 c \left (-4 b e g +c d g +7 c e f \right ) \sqrt {e x +d}}{4 e^{2} \left (-b e +2 c d \right )^{4} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {5 \, {\left (c^{2} d g + 7 \, c^{2} f e - 4 \, b c g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{3} d^{2} g + 2 \, c^{3} d f e - 3 \, b c^{2} d g e - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{2} d g - b c^{2} f e^{2} + b^{2} c g e^{2} - 9 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{2} f e + 6 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b c g e\right )}}{3 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} + \frac {10 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 26 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e + 3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e - 3 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 13 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2} - 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c g e^{2} + 11 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} f e - 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c g e}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left (x e + d\right )}^{2} c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________