14.189 Problem number 1873

\[ \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx \]

Optimal antiderivative \[ \frac {1}{3 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{3}}+\frac {c d}{2 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}+\frac {c^{2} d^{2}}{\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )}+\frac {c^{3} d^{3} \ln \left (c d x +a e \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{4}}-\frac {c^{3} d^{3} \ln \left (e x +d \right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{4}} \]

command

integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {c^{4} d^{4} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} - \frac {c^{3} d^{3} e \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac {11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (x e + d\right )}^{3}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________