\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{4 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {11}{2}}}+\frac {c^{3} \left (-8 b e g +11 c d g +5 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{64 e^{2} \left (-b e +2 c d \right )^{\frac {7}{2}}}-\frac {\left (-8 b e g +11 c d g +5 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{24 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {c \left (-8 b e g +11 c d g +5 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{96 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {c^{2} \left (-8 b e g +11 c d g +5 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{64 e^{2} \left (-b e +2 c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\frac {3 \, {\left (11 \, c^{5} d g + 5 \, c^{5} f e - 8 \, b c^{4} 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 )}{{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {-2 \, c d + b e}} + \frac {264 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{8} d^{4} g + 120 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{8} d^{3} f e - 588 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{7} d^{3} g e + 28 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{7} d^{3} g - 180 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{7} d^{2} f e^{2} + 486 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{6} d^{2} g e^{2} + 292 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{7} d^{2} f e - 188 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{6} d^{2} g e - 242 \, {\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^{6} d^{2} g + 90 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{6} d f e^{3} - 177 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{5} d g e^{3} - 292 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{6} d f e^{2} + 167 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{5} d g e^{2} - 110 \, {\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^{6} d f e + 297 \, {\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^{5} d g e - 33 \, {\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^{5} d g - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{5} f e^{4} + 24 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{4} c^{4} g e^{4} + 73 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{5} f e^{3} - 40 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{3} c^{4} g e^{3} + 55 \, {\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^{5} f e^{2} - 88 \, {\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^{2} c^{4} g e^{2} - 15 \, {\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^{5} f e + 24 \, {\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} b c^{4} g e}{{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (x e + d\right )}^{4} c^{4}}\right )} e^{\left (-2\right )}}{192 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \]________________________________________________________________________________________