\[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{c \,e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {4 \left (-4 b e g +5 c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 c^{3} e^{2} \sqrt {e x +d}}+\frac {2 \left (-4 b e g +5 c d g +3 c e f \right ) \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{3 c^{2} e^{2} \left (-b e +2 c d \right )} \]
command
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, c^{2} d^{2} g + 2 \, c^{2} d f e - 3 \, b c d g e - b c f e^{2} + b^{2} g e^{2}\right )} e^{\left (-2\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3}} - \frac {4 \, {\left (10 \, c^{2} d^{2} g + 6 \, c^{2} d f e - 13 \, b c d g e - 3 \, b c f e^{2} + 4 \, b^{2} g e^{2}\right )} e^{\left (-2\right )}}{3 \, \sqrt {2 \, c d - b e} c^{3}} + \frac {2 \, {\left (9 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{7} d g e^{4} + 3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{7} f e^{5} - 6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{6} g e^{5} - {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{6} g e^{4}\right )} e^{\left (-6\right )}}{3 \, c^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________