\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{2 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {c \left (-4 b e g +7 c d g +c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{4 e^{2} \left (-b e +2 c d \right )^{\frac {3}{2}}}-\frac {\left (-4 b e g +7 c d g +c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}} \]
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)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\frac {{\left (7 \, c^{3} d g + c^{3} f e - 4 \, b c^{2} g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {14 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g + 2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d f e - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} d g e - 9 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} f e^{2} + 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{2} g e^{2} + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} f e + 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} g e}{{\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-2\right )}}{4 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]________________________________________________________________________________________