\[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {-A \,e^{2}+B d e -C \,d^{2}}{2 e \left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{2}}+\frac {-2 A c d e -a B \,e^{2}+B c \,d^{2}+2 a C d e}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {\left (B c d \left (-3 a \,e^{2}+c \,d^{2}\right )-\left (A c -a C \right ) e \left (-a \,e^{2}+3 c \,d^{2}\right )\right ) \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}+\frac {\left (B c d \left (-3 a \,e^{2}+c \,d^{2}\right )-\left (A c -a C \right ) e \left (-a \,e^{2}+3 c \,d^{2}\right )\right ) \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3}}+\frac {\left (A c d \left (-3 a \,e^{2}+c \,d^{2}\right )-a \left (c \,d^{2} \left (-3 B e +C d \right )-a \,e^{2} \left (-B e +3 C d \right )\right )\right ) \arctan \left (\frac {x \sqrt {c}}{\sqrt {a}}\right ) \sqrt {c}}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a}} \]
command
integrate((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {C c^{2} d^{6} - 3 \, B c^{2} d^{5} e - 3 \, {\left (C a^{2} - 2 \, A a c\right )} d^{2} e^{4} - {\left ({\left (C a c - A c^{2}\right )} d^{5} e + B a^{2} x^{2} e^{6} + {\left (2 \, B a^{2} d x - 3 \, {\left (C a^{2} - A a c\right )} d x^{2}\right )} e^{5} - {\left (3 \, B a c d^{2} x^{2} - B a^{2} d^{2} + 6 \, {\left (C a^{2} - A a c\right )} d^{2} x\right )} e^{4} - {\left (6 \, B a c d^{3} x - {\left (C a c - A c^{2}\right )} d^{3} x^{2} + 3 \, {\left (C a^{2} - A a c\right )} d^{3}\right )} e^{3} - {\left (3 \, B a c d^{4} - 2 \, {\left (C a c - A c^{2}\right )} d^{4} x\right )} e^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + {\left (2 \, B a^{2} x + A a^{2}\right )} e^{6} + {\left (B a^{2} d - 4 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - 2 \, {\left (B a c d^{3} + 2 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} - {\left (2 \, B c^{2} d^{4} x + {\left (2 \, C a c - 5 \, A c^{2}\right )} d^{4}\right )} e^{2} - {\left (B c^{2} d^{5} e - {\left (C a^{2} - A a c\right )} x^{2} e^{6} - {\left (3 \, B a c d x^{2} + 2 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - {\left (6 \, B a c d^{2} x - 3 \, {\left (C a c - A c^{2}\right )} d^{2} x^{2} + {\left (C a^{2} - A a c\right )} d^{2}\right )} e^{4} + {\left (B c^{2} d^{3} x^{2} - 3 \, B a c d^{3} + 6 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} + {\left (2 \, B c^{2} d^{4} x + 3 \, {\left (C a c - A c^{2}\right )} d^{4}\right )} e^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B c^{2} d^{5} e - {\left (C a^{2} - A a c\right )} x^{2} e^{6} - {\left (3 \, B a c d x^{2} + 2 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - {\left (6 \, B a c d^{2} x - 3 \, {\left (C a c - A c^{2}\right )} d^{2} x^{2} + {\left (C a^{2} - A a c\right )} d^{2}\right )} e^{4} + {\left (B c^{2} d^{3} x^{2} - 3 \, B a c d^{3} + 6 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} + {\left (2 \, B c^{2} d^{4} x + 3 \, {\left (C a c - A c^{2}\right )} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (2 \, c^{3} d^{7} x e^{2} + c^{3} d^{8} e + 6 \, a c^{2} d^{5} x e^{4} + 6 \, a^{2} c d^{3} x e^{6} + a^{3} x^{2} e^{9} + 2 \, a^{3} d x e^{8} + {\left (3 \, a^{2} c d^{2} x^{2} + a^{3} d^{2}\right )} e^{7} + 3 \, {\left (a c^{2} d^{4} x^{2} + a^{2} c d^{4}\right )} e^{5} + {\left (c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6}\right )} e^{3}\right )}}, -\frac {C c^{2} d^{6} - 3 \, B c^{2} d^{5} e - 3 \, {\left (C a^{2} - 2 \, A a c\right )} d^{2} e^{4} + 2 \, {\left ({\left (C a c - A c^{2}\right )} d^{5} e + B a^{2} x^{2} e^{6} + {\left (2 \, B a^{2} d x - 3 \, {\left (C a^{2} - A a c\right )} d x^{2}\right )} e^{5} - {\left (3 \, B a c d^{2} x^{2} - B a^{2} d^{2} + 6 \, {\left (C a^{2} - A a c\right )} d^{2} x\right )} e^{4} - {\left (6 \, B a c d^{3} x - {\left (C a c - A c^{2}\right )} d^{3} x^{2} + 3 \, {\left (C a^{2} - A a c\right )} d^{3}\right )} e^{3} - {\left (3 \, B a c d^{4} - 2 \, {\left (C a c - A c^{2}\right )} d^{4} x\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + {\left (2 \, B a^{2} x + A a^{2}\right )} e^{6} + {\left (B a^{2} d - 4 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - 2 \, {\left (B a c d^{3} + 2 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} - {\left (2 \, B c^{2} d^{4} x + {\left (2 \, C a c - 5 \, A c^{2}\right )} d^{4}\right )} e^{2} - {\left (B c^{2} d^{5} e - {\left (C a^{2} - A a c\right )} x^{2} e^{6} - {\left (3 \, B a c d x^{2} + 2 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - {\left (6 \, B a c d^{2} x - 3 \, {\left (C a c - A c^{2}\right )} d^{2} x^{2} + {\left (C a^{2} - A a c\right )} d^{2}\right )} e^{4} + {\left (B c^{2} d^{3} x^{2} - 3 \, B a c d^{3} + 6 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} + {\left (2 \, B c^{2} d^{4} x + 3 \, {\left (C a c - A c^{2}\right )} d^{4}\right )} e^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B c^{2} d^{5} e - {\left (C a^{2} - A a c\right )} x^{2} e^{6} - {\left (3 \, B a c d x^{2} + 2 \, {\left (C a^{2} - A a c\right )} d x\right )} e^{5} - {\left (6 \, B a c d^{2} x - 3 \, {\left (C a c - A c^{2}\right )} d^{2} x^{2} + {\left (C a^{2} - A a c\right )} d^{2}\right )} e^{4} + {\left (B c^{2} d^{3} x^{2} - 3 \, B a c d^{3} + 6 \, {\left (C a c - A c^{2}\right )} d^{3} x\right )} e^{3} + {\left (2 \, B c^{2} d^{4} x + 3 \, {\left (C a c - A c^{2}\right )} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (2 \, c^{3} d^{7} x e^{2} + c^{3} d^{8} e + 6 \, a c^{2} d^{5} x e^{4} + 6 \, a^{2} c d^{3} x e^{6} + a^{3} x^{2} e^{9} + 2 \, a^{3} d x e^{8} + {\left (3 \, a^{2} c d^{2} x^{2} + a^{3} d^{2}\right )} e^{7} + 3 \, {\left (a c^{2} d^{4} x^{2} + a^{2} c d^{4}\right )} e^{5} + {\left (c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6}\right )} e^{3}\right )}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]