\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {105 c}{x^{8}}+\frac {120 d}{x^{7}}+\frac {140 e}{x^{6}}+\frac {168 f}{x^{5}}+\frac {210 g}{x^{4}}\right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{840}-\frac {b^{2} e \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {b \left (\frac {63 c}{x^{5}}+\frac {90 d}{x^{4}}+\frac {140 e}{x^{3}}+\frac {252 f}{x^{2}}+\frac {630 g}{x}\right ) \sqrt {b \,x^{3}+a}}{560}-\frac {27 b^{2} c \sqrt {b \,x^{3}+a}}{320 a \,x^{2}}-\frac {27 b^{2} d \sqrt {b \,x^{3}+a}}{112 a x}+\frac {27 b^{\frac {4}{3}} \left (14 a g +b d \right ) \sqrt {b \,x^{3}+a}}{112 a \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {27 \,3^{\frac {1}{4}} b^{\frac {4}{3}} \left (14 a g +b d \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{224 a^{\frac {2}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {9 \,3^{\frac {3}{4}} b^{\frac {4}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (7 b^{\frac {1}{3}} \left (-16 a f +b c \right )+20 a^{\frac {1}{3}} \left (14 a g +b d \right ) \left (1-\sqrt {3}\right )\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{2240 a \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^9,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {420 \, \sqrt {a} b^{2} x^{8} e \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 567 \, {\left (b^{2} c - 16 \, a b f\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 1620 \, {\left (b^{2} d + 14 \, a b g\right )} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (60 \, {\left (27 \, b^{2} d + 154 \, a b g\right )} x^{7} + 21 \, {\left (27 \, b^{2} c + 208 \, a b f\right )} x^{6} + 120 \, {\left (17 \, a b d + 14 \, a^{2} g\right )} x^{4} + 960 \, a^{2} d x + 84 \, {\left (19 \, a b c + 16 \, a^{2} f\right )} x^{3} + 840 \, a^{2} c + 560 \, {\left (5 \, a b x^{5} + 2 \, a^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{6720 \, a x^{8}}, \frac {840 \, \sqrt {-a} b^{2} x^{8} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) e - 567 \, {\left (b^{2} c - 16 \, a b f\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 1620 \, {\left (b^{2} d + 14 \, a b g\right )} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (60 \, {\left (27 \, b^{2} d + 154 \, a b g\right )} x^{7} + 21 \, {\left (27 \, b^{2} c + 208 \, a b f\right )} x^{6} + 120 \, {\left (17 \, a b d + 14 \, a^{2} g\right )} x^{4} + 960 \, a^{2} d x + 84 \, {\left (19 \, a b c + 16 \, a^{2} f\right )} x^{3} + 840 \, a^{2} c + 560 \, {\left (5 \, a b x^{5} + 2 \, a^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{6720 \, a x^{8}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{9}}, x\right ) \]