\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ \frac {5 c \left (4 b e g -11 c d g +3 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{12 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {\left (4 b e g -11 c d g +3 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{4 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{2 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {11}{2}}}-\frac {5 c \left (4 b e g -11 c d g +3 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right ) \sqrt {-b e +2 c d}}{4 e^{2}}+\frac {5 c \left (4 b e g -11 c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 e^{2} \sqrt {e x +d}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (120 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d g - 24 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} f e - 48 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} g e + 8 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} g + \frac {15 \, {\left (22 \, c^{4} d^{2} g - 6 \, c^{4} d f e - 19 \, b c^{3} d g e + 3 \, b c^{3} f e^{2} + 4 \, b^{2} c^{2} g e^{2}\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} d^{3} g - 28 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} d^{2} f e - 76 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{4} d^{2} g e - 34 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d^{2} g + 28 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{4} d f e^{2} + 31 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{3} d g e^{2} + 18 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d f e + 25 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} d g e - 7 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{3} f e^{3} - 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{2} g e^{3} - 9 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} f e^{2} - 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{2} g e^{2}\right )}}{{\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-2\right )}}{12 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________