\[ \int \frac {1}{x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 a^{2} d \,x^{2}}-\frac {c \left (c d \,x^{2}+a e \right )}{4 a^{2} \left (a \,e^{2}+c \,d^{2}\right ) \left (c \,x^{4}+a \right )}-\frac {c^{\frac {3}{2}} d \arctan \left (\frac {x^{2} \sqrt {c}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {c^{\frac {3}{2}} d \left (2 a \,e^{2}+c \,d^{2}\right ) \arctan \left (\frac {x^{2} \sqrt {c}}{\sqrt {a}}\right )}{2 a^{\frac {5}{2}} \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {e \ln \left (x \right )}{a^{2} d^{2}}+\frac {e^{5} \ln \left (e \,x^{2}+d \right )}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {c e \left (2 a \,e^{2}+c \,d^{2}\right ) \ln \left (c \,x^{4}+a \right )}{4 a^{2} \left (a \,e^{2}+c \,d^{2}\right )^{2}} \]
command
integrate(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {6 \, c^{3} d^{5} x^{4} + 2 \, a c^{2} d^{4} x^{2} e + 4 \, a c^{2} d^{5} + 2 \, a^{2} c d^{2} x^{2} e^{3} - 4 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} \log \left (x^{2} e + d\right ) - {\left (3 \, c^{3} d^{5} x^{6} + 3 \, a c^{2} d^{5} x^{2} + 5 \, {\left (a c^{2} d^{3} x^{6} + a^{2} c d^{3} x^{2}\right )} e^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 4 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{4} + 2 \, {\left (5 \, a c^{2} d^{3} x^{4} + 4 \, a^{2} c d^{3}\right )} e^{2} - 2 \, {\left (2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (c x^{4} + a\right ) + 8 \, {\left ({\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (x\right )}{8 \, {\left (a^{2} c^{3} d^{6} x^{6} + a^{3} c^{2} d^{6} x^{2} + {\left (a^{4} c d^{2} x^{6} + a^{5} d^{2} x^{2}\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{6} + a^{4} c d^{4} x^{2}\right )} e^{2}\right )}}, -\frac {3 \, c^{3} d^{5} x^{4} + a c^{2} d^{4} x^{2} e + 2 \, a c^{2} d^{5} + a^{2} c d^{2} x^{2} e^{3} - 2 \, {\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} \log \left (x^{2} e + d\right ) - {\left (3 \, c^{3} d^{5} x^{6} + 3 \, a c^{2} d^{5} x^{2} + 5 \, {\left (a c^{2} d^{3} x^{6} + a^{2} c d^{3} x^{2}\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) + 2 \, {\left (a^{2} c d x^{4} + a^{3} d\right )} e^{4} + {\left (5 \, a c^{2} d^{3} x^{4} + 4 \, a^{2} c d^{3}\right )} e^{2} - {\left (2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (c x^{4} + a\right ) + 4 \, {\left ({\left (a^{2} c x^{6} + a^{3} x^{2}\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{6} + a^{2} c d^{2} x^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{6} + a c^{2} d^{4} x^{2}\right )} e\right )} \log \left (x\right )}{4 \, {\left (a^{2} c^{3} d^{6} x^{6} + a^{3} c^{2} d^{6} x^{2} + {\left (a^{4} c d^{2} x^{6} + a^{5} d^{2} x^{2}\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{6} + a^{4} c d^{4} x^{2}\right )} e^{2}\right )}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]