\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx \]
Optimal antiderivative \[ \frac {b \left (-4 a g +b d \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{12 a^{\frac {3}{2}}}-\frac {\left (\frac {60 c}{x^{7}}+\frac {70 d}{x^{6}}+\frac {84 e}{x^{5}}+\frac {105 f}{x^{4}}+\frac {140 g}{x^{3}}\right ) \sqrt {b \,x^{3}+a}}{420}-\frac {3 b c \sqrt {b \,x^{3}+a}}{56 a \,x^{4}}-\frac {b d \sqrt {b \,x^{3}+a}}{12 a \,x^{3}}-\frac {3 b e \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}+\frac {3 b \left (-14 a f +5 b c \right ) \sqrt {b \,x^{3}+a}}{112 a^{2} x}-\frac {3 b^{\frac {4}{3}} \left (-14 a f +5 b c \right ) \sqrt {b \,x^{3}+a}}{112 a^{2} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {3 \,3^{\frac {1}{4}} b^{\frac {4}{3}} \left (-14 a f +5 b c \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 {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 {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 (28 a^{\frac {2}{3}} b^{\frac {1}{3}} e -5 \left (-14 a f +5 b c \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}}}}{560 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}}}} \]
command
integrate((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^8,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {252 \, a b^{\frac {3}{2}} x^{7} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 35 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {a} x^{7} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} - 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} - 280 \, a^{2} d x - 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} - 240 \, a^{2} c - 84 \, {\left (3 \, a b x^{5} + 4 \, a^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}, -\frac {252 \, a b^{\frac {3}{2}} x^{7} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 70 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {-a} x^{7} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} - 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} - 280 \, a^{2} d x - 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} - 240 \, a^{2} c - 84 \, {\left (3 \, a b x^{5} + 4 \, a^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}}, x\right ) \]