\[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {128 \left (-a e g +c d f \right )^{3} \left (2 a \,e^{2} g -c d \left (-3 d g +5 e f \right )\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3465 c^{5} d^{5} e \left (e x +d \right )^{\frac {3}{2}}}+\frac {32 \left (-a e g +c d f \right )^{2} \left (g x +f \right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{231 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {16 \left (-a e g +c d f \right ) \left (g x +f \right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{99 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 \left (g x +f \right )^{4} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{11 c d \left (e x +d \right )^{\frac {3}{2}}}+\frac {128 g \left (-a e g +c d f \right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{1155 c^{4} d^{4} e \sqrt {e x +d}} \]
command
integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),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 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}}{\sqrt {e x + d}}\,{d x} \]_______________________________________________________