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