\[ \int \frac {x \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx \]
Optimal antiderivative \[ -\frac {\ln \! \left (2^{\frac {1}{3}} c^{\frac {1}{3}} x +\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}\right ) \left (e +\frac {-b e +2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}}}{6 c^{\frac {2}{3}} \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}+\frac {\ln \! \left (2^{\frac {2}{3}} c^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} c^{\frac {1}{3}} x \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}+\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}\right ) \left (e +\frac {-b e +2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}}}{12 c^{\frac {2}{3}} \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}-\frac {\arctan \! \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} c^{\frac {1}{3}} x}{\left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \left (e +\frac {-b e +2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6 c^{\frac {2}{3}} \left (b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}-\frac {\ln \! \left (2^{\frac {1}{3}} c^{\frac {1}{3}} x +\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}\right ) \left (e +\frac {b e -2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}}}{6 c^{\frac {2}{3}} \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}+\frac {\ln \! \left (2^{\frac {2}{3}} c^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} c^{\frac {1}{3}} x \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}+\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {2}{3}}\right ) \left (e +\frac {b e -2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}}}{12 c^{\frac {2}{3}} \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}-\frac {\arctan \! \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} c^{\frac {1}{3}} x}{\left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \left (e +\frac {b e -2 c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6 c^{\frac {2}{3}} \left (b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{3}}} \]
command
integrate(x*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {Timed out} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {output too large to display} \]