\[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{\frac {11}{2}}}{3 c \,e^{2} \left (-b e +2 c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {16 \left (-b e +2 c d \right ) \left (-8 b e g +11 c d g +5 c e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{15 c^{4} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {4 \left (-8 b e g +11 c d g +5 c e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{15 c^{3} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {2 \left (-8 b e g +11 c d g +5 c e f \right ) \left (e x +d \right )^{\frac {7}{2}}}{15 c^{2} e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}-\frac {32 \left (-b e +2 c d \right )^{2} \left (-8 b e g +11 c d g +5 c e f \right ) \sqrt {e x +d}}{15 c^{5} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((e*x+d)^(11/2)*(g*x+f)/(-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 {32 \, {\left (44 \, c^{3} d^{3} g + 20 \, c^{3} d^{2} f e - 76 \, b c^{2} d^{2} g e - 20 \, b c^{2} d f e^{2} + 43 \, b^{2} c d g e^{2} + 5 \, b^{2} c f e^{3} - 8 \, b^{3} g e^{3}\right )} e^{\left (-2\right )}}{15 \, \sqrt {2 \, c d - b e} c^{5}} - \frac {2 \, {\left (8 \, c^{4} d^{4} g + 8 \, c^{4} d^{3} f e - 20 \, b c^{3} d^{3} g e + 60 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{3} d^{3} g - 12 \, b c^{3} d^{2} f e^{2} + 18 \, b^{2} c^{2} d^{2} g e^{2} + 36 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{3} d^{2} f e - 108 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b c^{2} d^{2} g e + 6 \, b^{2} c^{2} d f e^{3} - 7 \, b^{3} c d g e^{3} - 36 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b c^{2} d f e^{2} + 63 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b^{2} c d g e^{2} - b^{3} c f e^{4} + b^{4} g e^{4} + 9 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b^{2} c f e^{3} - 12 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b^{3} g e^{3}\right )} e^{\left (-2\right )}}{3 \, {\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} c^{5}} - \frac {2 \, {\left (270 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{22} d^{2} g e^{8} + 90 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{22} d f e^{9} - 315 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{21} d g e^{9} - 35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{21} d g e^{8} - 45 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{21} f e^{10} + 90 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{20} g e^{10} - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{21} f e^{9} + 20 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{20} g e^{9} + 3 \, {\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^{20} g e^{8}\right )} e^{\left (-10\right )}}{15 \, c^{25}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________