\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {12 c}{x^{5}}+\frac {15 d}{x^{4}}+\frac {20 e}{x^{3}}+\frac {30 f}{x^{2}}+\frac {60 g}{x}\right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{60}-b e \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) \sqrt {a}+\frac {27 b c \sqrt {b \,x^{3}+a}}{20 x^{2}}-\frac {27 b d \sqrt {b \,x^{3}+a}}{8 x}-\frac {b \left (-180 g \,x^{5}-126 f \,x^{4}-140 e \,x^{3}-315 d \,x^{2}+252 c x \right ) \sqrt {b \,x^{3}+a}}{140 x^{3}}+\frac {27 b^{\frac {1}{3}} \left (8 a g +7 b d \right ) \sqrt {b \,x^{3}+a}}{56 \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {27 \,3^{\frac {1}{4}} a^{\frac {1}{3}} b^{\frac {1}{3}} \left (8 a g +7 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}}}}{112 \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 {1}{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 (14 b^{\frac {1}{3}} \left (2 a f +b c \right )-5 a^{\frac {1}{3}} \left (8 a g +7 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}}}}{280 \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^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {210 \, \sqrt {a} b x^{5} e \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1134 \, {\left (b c + 2 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 405 \, {\left (7 \, b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (240 \, b g x^{7} + 336 \, b f x^{6} - 105 \, {\left (11 \, b d + 8 \, a g\right )} x^{4} - 42 \, {\left (13 \, b c + 10 \, a f\right )} x^{3} - 210 \, a d x - 168 \, a c + 280 \, {\left (2 \, b x^{5} - a x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{840 \, x^{5}}, \frac {420 \, \sqrt {-a} b x^{5} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) e + 1134 \, {\left (b c + 2 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 405 \, {\left (7 \, b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (240 \, b g x^{7} + 336 \, b f x^{6} - 105 \, {\left (11 \, b d + 8 \, a g\right )} x^{4} - 42 \, {\left (13 \, b c + 10 \, a f\right )} x^{3} - 210 \, a d x - 168 \, a c + 280 \, {\left (2 \, b x^{5} - a x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{840 \, x^{5}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{6}}, x\right ) \]