\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 g \left (e x +d \right )^{\frac {3}{2}} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{5 c \,e^{2}}-\frac {4 \left (-b e +2 c d \right ) \left (-4 b e g +3 c d g +5 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{15 c^{3} e^{2} \sqrt {e x +d}}-\frac {2 \left (-4 b e g +3 c d g +5 c e f \right ) \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{15 c^{2} e^{2}} \]
command
integrate((e*x+d)^(3/2)*(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
\[ -\frac {2}{15} \, {\left (\frac {15 \, {\left (2 \, c^{2} d^{2} g + 2 \, c^{2} d f e - 3 \, b c d g e - b c f e^{2} + b^{2} g e^{2}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} e^{\left (-1\right )}}{c^{3}} - \frac {2 \, {\left (6 \, \sqrt {2 \, c d - b e} c^{2} d^{2} g + 10 \, \sqrt {2 \, c d - b e} c^{2} d f e - 11 \, \sqrt {2 \, c d - b e} b c d g e - 5 \, \sqrt {2 \, c d - b e} b c f e^{2} + 4 \, \sqrt {2 \, c d - b e} b^{2} g e^{2}\right )} e^{\left (-1\right )}}{c^{3}} - \frac {{\left (15 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d g + 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c f e - 10 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b g e - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} g\right )} e^{\left (-1\right )}}{c^{3}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________