\[ \int \frac {x^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {f x}{b^{2}}+\frac {g \,x^{2}}{2 b^{2}}+\frac {h \,x^{3}}{3 b^{2}}-\frac {x \left (b c -a f +\left (-a g +b d \right ) x +\left (-a h +b e \right ) x^{2}\right )}{3 b^{2} \left (b \,x^{3}+a \right )}+\frac {\left (b^{\frac {1}{3}} \left (-4 a f +b c \right )-a^{\frac {1}{3}} \left (-5 a g +2 b d \right )\right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {2}{3}} b^{\frac {8}{3}}}-\frac {\left (b^{\frac {1}{3}} \left (-4 a f +b c \right )-a^{\frac {1}{3}} \left (-5 a g +2 b d \right )\right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{18 a^{\frac {2}{3}} b^{\frac {8}{3}}}+\frac {\left (-2 a h +b e \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {\left (b^{\frac {4}{3}} c +2 a^{\frac {1}{3}} b d -4 a \,b^{\frac {1}{3}} f -5 a^{\frac {4}{3}} g \right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {2}{3}} b^{\frac {8}{3}}} \]
command
integrate(x^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]