\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^4} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (315 g \,x^{5}+385 f \,x^{4}+495 e \,x^{3}+693 d \,x^{2}+1155 c x \right )}{3465 x^{4}}-\frac {\left (2 a f +3 b c \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {a c \sqrt {b \,x^{3}+a}}{x^{3}}+\frac {27 a d \sqrt {b \,x^{3}+a}}{10 x^{2}}-\frac {27 a e \sqrt {b \,x^{3}+a}}{7 x}-\frac {2 a \left (-189 g \,x^{5}-385 f \,x^{4}-1485 e \,x^{3}+2079 d \,x^{2}+1155 c x \right ) \sqrt {b \,x^{3}+a}}{1155 x^{4}}+\frac {27 a \,b^{\frac {1}{3}} e \sqrt {b \,x^{3}+a}}{7 \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {27 \,3^{\frac {1}{4}} a^{\frac {4}{3}} b^{\frac {1}{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}}}}{14 \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}} a \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 (77 b d +28 a g -110 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}}}}{770 b^{\frac {1}{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((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {53460 \, a b^{\frac {3}{2}} x^{3} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 1155 \, {\left (3 \, b^{2} c + 2 \, a b f\right )} \sqrt {a} x^{3} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 3402 \, {\left (11 \, a b d + 4 \, a^{2} g\right )} \sqrt {b} x^{3} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 2 \, {\left (1260 \, b^{2} g x^{7} + 1540 \, b^{2} f x^{6} + 252 \, {\left (11 \, b^{2} d + 14 \, a b g\right )} x^{4} - 3465 \, a b d x + 1540 \, {\left (3 \, b^{2} c + 4 \, a b f\right )} x^{3} - 2310 \, a b c + 990 \, {\left (2 \, b^{2} x^{5} - 7 \, a b x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{13860 \, b x^{3}}, -\frac {26730 \, a b^{\frac {3}{2}} x^{3} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 1155 \, {\left (3 \, b^{2} c + 2 \, a b f\right )} \sqrt {-a} x^{3} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 1701 \, {\left (11 \, a b d + 4 \, a^{2} g\right )} \sqrt {b} x^{3} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left (1260 \, b^{2} g x^{7} + 1540 \, b^{2} f x^{6} + 252 \, {\left (11 \, b^{2} d + 14 \, a b g\right )} x^{4} - 3465 \, a b d x + 1540 \, {\left (3 \, b^{2} c + 4 \, a b f\right )} x^{3} - 2310 \, a b c + 990 \, {\left (2 \, b^{2} x^{5} - 7 \, a b x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{6930 \, b x^{3}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{4}}, x\right ) \]