\[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (-b e +2 c d \right ) \left (-4 b e g +c d g +7 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{105 c^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-4 b e g +c d g +7 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{35 c^{2} e^{2} \sqrt {e x +d}}-\frac {2 g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}{7 c \,e^{2}} \]
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)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{105} \, {\left (7 \, d g {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b 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}}{c^{2}}\right )} e^{\left (-1\right )} + 35 \, d f {\left (\frac {{\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}}{c} - \frac {2 \, \sqrt {2 \, c d - b e} c d - \sqrt {2 \, c d - b e} b e}{c}\right )} - g {\left (\frac {{\left (22 \, \sqrt {2 \, c d - b e} c^{3} d^{3} - 19 \, \sqrt {2 \, c d - b e} b c^{2} d^{2} e + 20 \, \sqrt {2 \, c d - b e} b^{2} c d e^{2} - 8 \, \sqrt {2 \, c d - b e} b^{3} e^{3}\right )} e^{\left (-2\right )}}{c^{3}} - \frac {{\left (35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d^{2} - 70 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c d e - 42 \, {\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 + 35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} e^{2} + 42 \, {\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 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}\right )} e^{\left (-2\right )}}{c^{3}}\right )} e + 7 \, f {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b 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}}{c^{2}}\right )}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________