\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{c d \left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {2 \left (2 a \,e^{2} g -c d \left (d g +e f \right )\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{c^{2} d^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}} \]
command
integrate((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{c^{2} d^{2}} + \frac {2 \, {\left (c d^{2} g + c d f e - 2 \, a g e^{2}\right )}}{\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{2}} - \frac {2 \, {\left (c d f e - a g e^{2}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________