\[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-2 b e g -c d g +5 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{e^{2} \left (-b e +2 c d \right )^{\frac {7}{2}}}+\frac {2 \left (-b e g +c d g +c e f \right ) \sqrt {e x +d}}{3 c \,e^{2} \left (-b e +2 c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {2 b e g +c d g -5 c e f}{3 c \,e^{2} \left (-b e +2 c d \right )^{2} \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {\left (-2 b e g -c d g +5 c e f \right ) \sqrt {e x +d}}{e^{2} \left (-b e +2 c d \right )^{3} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((e*x+d)^(1/2)*(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
\[ -\frac {{\left (c d g - 5 \, c f e + 2 \, b g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{2} d^{2} g + 2 \, c^{2} d f e - 3 \, b c d g e - b c f e^{2} + b^{2} g e^{2} - 6 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c f e + 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b g e\right )}}{3 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c f e}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left (x e + d\right )} c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________