\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {16 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{693 c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}+\frac {8 \left (-a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{99 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{11 c d \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{\sqrt {e x + d}}\,{d x} \]_______________________________________________________