\[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}}}{3 \left (-a e g +c d f \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {2 g^{\frac {3}{2}} \arctan \left (\frac {\sqrt {g}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a e g +c d f}\, \sqrt {e x +d}}\right )}{\left (-a e g +c d f \right )^{\frac {5}{2}}}+\frac {2 g \sqrt {e x +d}}{\left (-a e g +c d f \right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate((e*x+d)^(5/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{3} \, {\left (\frac {3 \, g^{2} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{\left (-1\right )}}{{\left (c^{2} d^{2} f^{2} e - 2 \, a c d f g e^{2} + a^{2} g^{2} e^{3}\right )} \sqrt {c d f g - a g^{2} e}} - \frac {c d f e^{2} - a g e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} g}{{\left (c^{2} d^{2} f^{2} e - 2 \, a c d f g e^{2} + a^{2} g^{2} e^{3}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}\right )} e^{2} - \frac {2 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} c d^{2} g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} a g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{2} + 3 \, \sqrt {c d f g - a g^{2} e} c d^{2} g e + \sqrt {c d f g - a g^{2} e} c d f e^{2} - 4 \, \sqrt {c d f g - a g^{2} e} a g e^{3}\right )}}{3 \, {\left (\sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} f^{2} - 2 \, \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} f g e - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} f^{2} e^{2} + \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} g^{2} e^{2} + 2 \, \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{2} c d f g e^{3} - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{3} g^{2} e^{4}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________