\[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{\frac {9}{2}}}{c \,e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {12 \left (-8 b e g +9 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{35 c^{3} e^{2}}+\frac {2 \left (-8 b e g +9 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {5}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{7 c^{2} e^{2} \left (-b e +2 c d \right )}+\frac {32 \left (-b e +2 c d \right )^{2} \left (-8 b e g +9 c d g +7 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{35 c^{5} e^{2} \sqrt {e x +d}}+\frac {16 \left (-b e +2 c d \right ) \left (-8 b e g +9 c d g +7 c e f \right ) \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{35 c^{4} e^{2}} \]
command
integrate((e*x+d)^(9/2)*(g*x+f)/(-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 {2 \, {\left (8 \, c^{4} d^{4} g + 8 \, c^{4} d^{3} f e - 20 \, b c^{3} d^{3} g e - 12 \, b c^{3} d^{2} f e^{2} + 18 \, b^{2} c^{2} d^{2} g e^{2} + 6 \, b^{2} c^{2} d f e^{3} - 7 \, b^{3} c d g e^{3} - b^{3} c f e^{4} + b^{4} g e^{4}\right )} e^{\left (-2\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5}} - \frac {32 \, {\left (72 \, c^{4} d^{4} g + 56 \, c^{4} d^{3} f e - 172 \, b c^{3} d^{3} g e - 84 \, b c^{3} d^{2} f e^{2} + 150 \, b^{2} c^{2} d^{2} g e^{2} + 42 \, b^{2} c^{2} d f e^{3} - 57 \, b^{3} c d g e^{3} - 7 \, b^{3} c f e^{4} + 8 \, b^{4} g e^{4}\right )} e^{\left (-2\right )}}{35 \, \sqrt {2 \, c d - b e} c^{5}} + \frac {2 \, {\left (700 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{33} d^{3} g e^{12} + 420 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{33} d^{2} f e^{13} - 1260 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{32} d^{2} g e^{13} - 210 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{32} d^{2} g e^{12} - 420 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{32} d f e^{14} + 735 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{31} d g e^{14} - 70 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{32} d f e^{13} + 245 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{31} d g e^{13} + 49 \, {\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^{31} d g e^{12} + 105 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{31} f e^{15} - 140 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{30} g e^{15} + 35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{31} f e^{14} - 70 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{30} g e^{14} + 7 \, {\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^{31} f e^{13} - 28 \, {\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^{30} g e^{13} + 5 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{30} g e^{12}\right )} e^{\left (-14\right )}}{35 \, c^{35}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________