\[ \int \frac {(d+e x) (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (d \left (-b e +2 c d \right )+e \left (-b e +2 c d \right ) x \right )}{3 c \,e^{2} \left (-b e +2 c d \right )^{2} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {2 \left (-b e g -2 c d g +4 c e f \right ) \left (2 c x +b \right )}{3 c e \left (-b e +2 c d \right )^{3} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((e*x+d)*(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
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {2 \, {\left (4 \, c^{3} d^{2} g e^{3} - 8 \, c^{3} d f e^{4} + 4 \, b c^{2} f e^{5} - b^{2} c g e^{5}\right )} x}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}} + \frac {3 \, {\left (4 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 4 \, b^{2} c f e^{5} - b^{3} g e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac {3 \, {\left (8 \, c^{3} d^{3} f e^{2} - 4 \, b c^{2} d^{3} g e^{2} - 12 \, b c^{2} d^{2} f e^{3} + 8 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4} + b^{3} f e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac {8 \, c^{3} d^{5} g + 8 \, c^{3} d^{4} f e - 24 \, b c^{2} d^{4} g e - 4 \, b c^{2} d^{3} f e^{2} + 22 \, b^{2} c d^{3} g e^{2} - 6 \, b^{2} c d^{2} f e^{3} - 6 \, b^{3} d^{2} g e^{3} + 3 \, b^{3} d f e^{4}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )}}{3 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \]