\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-2 b e g -c d g +5 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{15 c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{5 c \,e^{2} \sqrt {e x +d}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{15} \, {\left (g {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b e - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{c^{2}}\right )} e^{\left (-1\right )} + 5 \, f {\left (\frac {{\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}}{c} - \frac {2 \, \sqrt {2 \, c d - b e} c d - \sqrt {2 \, c d - b e} b e}{c}\right )}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{\sqrt {e x + d}}\,{d x} \]________________________________________________________________________________________