\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 b e g -7 c d g +3 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {\left (2 b e g -7 c d g +3 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right ) \sqrt {-b e +2 c d}}{e^{2}}-\frac {\left (2 b e g -7 c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \sqrt {e x +d}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (18 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} d g - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{2} f e - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c g e + 2 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c g + \frac {3 \, {\left (14 \, c^{3} d^{2} g - 6 \, c^{3} d f e - 11 \, b c^{2} d g e + 3 \, b c^{2} f e^{2} + 2 \, b^{2} c g e^{2}\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e + \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2}\right )}}{{\left (x e + d\right )} c}\right )} e^{\left (-2\right )}}{3 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________