\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {b e \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\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 ) \sqrt {b \,x^{3}+a}}{60}-\frac {3 b c \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}-\frac {3 b d \sqrt {b \,x^{3}+a}}{8 a x}+\frac {3 b^{\frac {1}{3}} \left (8 a g +b d \right ) \sqrt {b \,x^{3}+a}}{8 a \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {3 \,3^{\frac {1}{4}} b^{\frac {1}{3}} \left (8 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}}}}{16 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 {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 (2 b^{\frac {1}{3}} \left (-10 a f +b c \right )+5 a^{\frac {1}{3}} \left (8 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}}}}{40 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((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {10 \, \sqrt {a} b x^{5} e \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 18 \, {\left (b c - 10 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 45 \, {\left (b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (15 \, {\left (3 \, b d + 8 \, a g\right )} x^{4} + 6 \, {\left (3 \, b c + 10 \, a f\right )} x^{3} + 40 \, a x^{2} e + 30 \, a d x + 24 \, a c\right )} \sqrt {b x^{3} + a}}{120 \, a x^{5}}, \frac {20 \, \sqrt {-a} b x^{5} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) e - 18 \, {\left (b c - 10 \, a f\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 45 \, {\left (b d + 8 \, a g\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (15 \, {\left (3 \, b d + 8 \, a g\right )} x^{4} + 6 \, {\left (3 \, b c + 10 \, a f\right )} x^{3} + 40 \, a x^{2} e + 30 \, a d x + 24 \, a c\right )} \sqrt {b x^{3} + a}}{120 \, a x^{5}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{6}}, x\right ) \]