\[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \]
Optimal antiderivative \[ \frac {256 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3465 c^{5} d^{5} \left (e x +d \right )^{\frac {3}{2}}}+\frac {16 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{99 c^{2} d^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{11 c d}+\frac {128 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{1155 c^{4} d^{4} \sqrt {e x +d}}+\frac {32 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}{231 c^{3} d^{3}} \]
command
integrate((e*x+d)^(7/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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 \[ \int \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {7}{2}}\,{d x} \]_______________________________________________________