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