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