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