\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}+\frac {c^{2} \left (-2 b e g +3 c d g +c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{8 e^{2} \left (-b e +2 c d \right )^{\frac {5}{2}}}-\frac {\left (-2 b e g +3 c d g +c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {c \left (-2 b e g +3 c d g +c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{8 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
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)^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\frac {3 \, {\left (3 \, c^{4} d g + c^{4} f e - 2 \, b c^{3} g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-2 \, c d + b e}} + \frac {36 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d^{3} g + 12 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d^{2} f e - 60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} d^{2} g e - 16 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d^{2} g - 12 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} d f e^{2} + 33 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{4} d g e^{2} + 16 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d f e + 8 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} d g e - 9 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d g + 3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{4} f e^{3} - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{3} g e^{3} - 8 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} f e^{2} - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} f e + 6 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} g e}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{3} c^{3}}\right )} e^{\left (-2\right )}}{24 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]________________________________________________________________________________________