\[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \]
Optimal antiderivative \[ \frac {128 \left (-a e g +c d f \right )^{3} \left (10 a \,e^{2} g +c d \left (-11 d g +e f \right )\right ) \left (2 a \,e^{2} g -c d \left (-d g +3 e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3465 c^{6} d^{6} e g \sqrt {e x +d}}-\frac {32 \left (-a e g +c d f \right )^{2} \left (10 a \,e^{2} g +c d \left (-11 d g +e f \right )\right ) \left (g x +f \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{1155 c^{4} d^{4} g \sqrt {e x +d}}-\frac {16 \left (-a e g +c d f \right ) \left (10 a \,e^{2} g +c d \left (-11 d g +e f \right )\right ) \left (g x +f \right )^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{693 c^{3} d^{3} g \sqrt {e x +d}}-\frac {2 \left (10 a \,e^{2} g +c d \left (-11 d g +e f \right )\right ) \left (g x +f \right )^{4} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{99 c^{2} d^{2} g \sqrt {e x +d}}+\frac {2 e \left (g x +f \right )^{5} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{11 c d g \sqrt {e x +d}}-\frac {128 \left (-a e g +c d f \right )^{3} \left (10 a \,e^{2} g +c d \left (-11 d g +e f \right )\right ) \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3465 c^{5} d^{5} e} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)^4/(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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{4}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]_______________________________________________________