\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {280 c}{x^{9}}+\frac {315 d}{x^{8}}+\frac {360 e}{x^{7}}+\frac {420 f}{x^{6}}+\frac {504 g}{x^{5}}\right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{2520}+\frac {b^{2} \left (-6 a f +b c \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {3}{2}}}-\frac {b \left (\frac {140 c}{x^{6}}+\frac {189 d}{x^{5}}+\frac {270 e}{x^{4}}+\frac {420 f}{x^{3}}+\frac {756 g}{x^{2}}\right ) \sqrt {b \,x^{3}+a}}{1680}-\frac {b^{2} c \sqrt {b \,x^{3}+a}}{24 a \,x^{3}}-\frac {27 b^{2} d \sqrt {b \,x^{3}+a}}{320 a \,x^{2}}-\frac {27 b^{2} e \sqrt {b \,x^{3}+a}}{112 a x}+\frac {27 b^{\frac {7}{3}} e \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 {7}{3}} e \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 {5}{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 d -112 a g +20 a^{\frac {1}{3}} b^{\frac {2}{3}} e \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^10,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {4860 \, a b^{\frac {5}{2}} x^{9} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 210 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {a} x^{9} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3} + 180 \, {\left (27 \, a b^{2} x^{8} + 34 \, a^{2} b x^{5} + 16 \, a^{3} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}, -\frac {4860 \, a b^{\frac {5}{2}} x^{9} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 420 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {-a} x^{9} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3} + 180 \, {\left (27 \, a b^{2} x^{8} + 34 \, a^{2} b x^{5} + 16 \, a^{3} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{10}}, x\right ) \]