\[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-6 b e g +5 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{35 c^{2} e^{2}}-\frac {2 g \left (e x +d \right )^{\frac {5}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{7 c \,e^{2}}-\frac {16 \left (-b e +2 c d \right )^{2} \left (-6 b e g +5 c d g +7 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{105 c^{4} e^{2} \sqrt {e x +d}}-\frac {8 \left (-b e +2 c d \right ) \left (-6 b e g +5 c d g +7 c e f \right ) \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{105 c^{3} e^{2}} \]
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)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{105} \, {\left (\frac {105 \, {\left (4 \, c^{3} d^{3} g + 4 \, c^{3} d^{2} f e - 8 \, b c^{2} d^{2} g e - 4 \, b c^{2} d f e^{2} + 5 \, b^{2} c d g e^{2} + b^{2} c f e^{3} - b^{3} g e^{3}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} e^{\left (-1\right )}}{c^{4}} - \frac {8 \, {\left (20 \, \sqrt {2 \, c d - b e} c^{3} d^{3} g + 28 \, \sqrt {2 \, c d - b e} c^{3} d^{2} f e - 44 \, \sqrt {2 \, c d - b e} b c^{2} d^{2} g e - 28 \, \sqrt {2 \, c d - b e} b c^{2} d f e^{2} + 29 \, \sqrt {2 \, c d - b e} b^{2} c d g e^{2} + 7 \, \sqrt {2 \, c d - b e} b^{2} c f e^{3} - 6 \, \sqrt {2 \, c d - b e} b^{3} g e^{3}\right )} e^{\left (-1\right )}}{c^{4}} - \frac {{\left (280 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d^{2} g + 140 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d f e - 350 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c d g e - 105 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c d g - 70 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c f e^{2} + 105 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} g e^{2} - 21 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c f e + 63 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b g e - 15 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} g\right )} e^{\left (-1\right )}}{c^{4}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________