\[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {16 \left (-a e g +c d f \right )^{2} \left (2 a \,e^{2} g -c d \left (-5 d g +7 e f \right )\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{1155 c^{4} d^{4} e \left (e x +d \right )^{\frac {5}{2}}}+\frac {16 g \left (-a e g +c d f \right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{231 c^{3} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 \left (-a e g +c d f \right ) \left (g x +f \right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{33 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 \left (g x +f \right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{11 c d \left (e x +d \right )^{\frac {5}{2}}} \]
command
integrate((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]_______________________________________________________