\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^{9}}+\frac {3465 f}{x^{8}}+\frac {3960 g}{x^{7}}\right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{27720}+\frac {b^{3} e \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {3}{2}}}-\frac {b \left (\frac {945 c}{x^{8}}+\frac {1188 d}{x^{7}}+\frac {1540 e}{x^{6}}+\frac {2079 f}{x^{5}}+\frac {2970 g}{x^{4}}\right ) \sqrt {b \,x^{3}+a}}{18480}-\frac {27 b^{2} c \sqrt {b \,x^{3}+a}}{1760 a \,x^{5}}-\frac {27 b^{2} d \sqrt {b \,x^{3}+a}}{1120 a \,x^{4}}-\frac {b^{2} e \sqrt {b \,x^{3}+a}}{24 a \,x^{3}}+\frac {27 b^{2} \left (-22 a f +7 b c \right ) \sqrt {b \,x^{3}+a}}{7040 a^{2} x^{2}}+\frac {27 b^{2} \left (-4 a g +b d \right ) \sqrt {b \,x^{3}+a}}{448 a^{2} x}-\frac {27 b^{\frac {7}{3}} \left (-4 a g +b d \right ) \sqrt {b \,x^{3}+a}}{448 a^{2} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {27 \,3^{\frac {1}{4}} b^{\frac {7}{3}} \left (-4 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}}}}{896 a^{\frac {5}{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 {7}{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 (-22 a f +7 b c \right )+110 a^{\frac {1}{3}} \left (-4 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}}}}{49280 a^{2} \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^12,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {4620 \, \sqrt {a} b^{3} x^{11} 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 ) + 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} \sqrt {b} x^{11} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} \sqrt {b} x^{11} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} x^{10} + 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} x^{9} - 396 \, {\left (27 \, a b^{2} d + 340 \, a^{2} b g\right )} x^{7} - 252 \, {\left (27 \, a b^{2} c + 418 \, a^{2} b f\right )} x^{6} - 44352 \, a^{3} d x - 3168 \, {\left (23 \, a^{2} b d + 20 \, a^{3} g\right )} x^{4} - 40320 \, a^{3} c - 2520 \, {\left (25 \, a^{2} b c + 22 \, a^{3} f\right )} x^{3} - 6160 \, {\left (3 \, a b^{2} x^{8} + 14 \, a^{2} b x^{5} + 8 \, a^{3} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{443520 \, a^{2} x^{11}}, -\frac {9240 \, \sqrt {-a} b^{3} x^{11} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) e - 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} \sqrt {b} x^{11} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} \sqrt {b} x^{11} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} x^{10} + 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} x^{9} - 396 \, {\left (27 \, a b^{2} d + 340 \, a^{2} b g\right )} x^{7} - 252 \, {\left (27 \, a b^{2} c + 418 \, a^{2} b f\right )} x^{6} - 44352 \, a^{3} d x - 3168 \, {\left (23 \, a^{2} b d + 20 \, a^{3} g\right )} x^{4} - 40320 \, a^{3} c - 2520 \, {\left (25 \, a^{2} b c + 22 \, a^{3} f\right )} x^{3} - 6160 \, {\left (3 \, a b^{2} x^{8} + 14 \, a^{2} b x^{5} + 8 \, a^{3} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{443520 \, a^{2} x^{11}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{12}}, x\right ) \]