\[ \int \frac {f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 c^{2} \left (-6 b e g +5 c d g +7 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{8 e^{2} \left (-b e +2 c d \right )^{\frac {9}{2}}}+\frac {d g -e f}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {5}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {6 b e g -5 c d g -7 c e f}{12 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}-\frac {5 c \left (-6 b e g +5 c d g +7 c e f \right )}{24 e^{2} \left (-b e +2 c d \right )^{3} \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {5 c^{2} \left (-6 b e g +5 c d g +7 c e f \right ) \sqrt {e x +d}}{8 e^{2} \left (-b e +2 c d \right )^{4} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^(5/2)/(-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 {5 \, {\left (5 \, c^{3} d g + 7 \, c^{3} f e - 6 \, b c^{2} 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 )}{8 \, {\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 (c^{3} d g + c^{3} f e - b c^{2} g e\right )}}{{\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 {-{\left (x e + d\right )} c + 2 \, c d - b e}} - \frac {84 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} d^{3} g + 348 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} d^{2} f e - 300 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{4} d^{2} g e - 112 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d^{2} g - 348 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{4} d f e^{2} + 237 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{3} d g e^{2} - 272 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d f e + 248 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} d g e + 27 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d g + 87 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{3} f e^{3} - 54 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{2} g e^{3} + 136 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} f e^{2} - 96 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{2} g e^{2} + 57 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} f e - 42 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} g e}{24 \, {\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 )}^{3} c^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________