\[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {32 \left (-b e +2 c d \right )^{3} \left (-8 b e g +c d g +15 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{45045 c^{5} e^{2} \left (e x +d \right )^{\frac {7}{2}}}-\frac {16 \left (-b e +2 c d \right )^{2} \left (-8 b e g +c d g +15 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{6435 c^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 \left (-b e +2 c d \right ) \left (-8 b e g +c d g +15 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{715 c^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-8 b e g +c d g +15 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{195 c^{2} e^{2} \sqrt {e x +d}}-\frac {2 g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}} \sqrt {e x +d}}{15 c \,e^{2}} \]
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
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \int {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} \sqrt {e x + d} {\left (g x + f\right )}\,{d x} \]_______________________________________________________