\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (6435 g \,x^{5}+7293 f \,x^{4}+8415 e \,x^{3}+9945 d \,x^{2}+12155 c x \right )}{109395 x}-\frac {2 a^{\frac {3}{2}} c \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {2 a^{2} f \sqrt {b \,x^{3}+a}}{15 b}+\frac {54 a^{2} g x \sqrt {b \,x^{3}+a}}{935 b}+\frac {2 a \left (12285 g \,x^{5}+17017 f \,x^{4}+25245 e \,x^{3}+41769 d \,x^{2}+85085 c x \right ) \sqrt {b \,x^{3}+a}}{255255 x}+\frac {54 a^{2} e \sqrt {b \,x^{3}+a}}{91 b^{\frac {2}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {27 \,3^{\frac {1}{4}} a^{\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}}}}{91 b^{\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 {18 \,3^{\frac {3}{4}} a^{2} \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 (1547 b d -182 a g -935 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}}}}{85085 b^{\frac {4}{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,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {255255 \, a^{\frac {3}{2}} b^{2} c \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 908820 \, a^{2} b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 88452 \, {\left (17 \, a^{2} b d - 2 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 4 \, {\left (45045 \, b^{3} g x^{7} + 51051 \, b^{3} f x^{6} + 4095 \, {\left (17 \, b^{3} d + 20 \, a b^{2} g\right )} x^{4} + 340340 \, a b^{2} c + 51051 \, a^{2} b f + 17017 \, {\left (5 \, b^{3} c + 6 \, a b^{2} f\right )} x^{3} + 819 \, {\left (238 \, a b^{2} d + 27 \, a^{2} b g\right )} x + 8415 \, {\left (7 \, b^{3} x^{5} + 16 \, a b^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{1531530 \, b^{2}}, \frac {255255 \, \sqrt {-a} a b^{2} c \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 454410 \, a^{2} b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 44226 \, {\left (17 \, a^{2} b d - 2 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (45045 \, b^{3} g x^{7} + 51051 \, b^{3} f x^{6} + 4095 \, {\left (17 \, b^{3} d + 20 \, a b^{2} g\right )} x^{4} + 340340 \, a b^{2} c + 51051 \, a^{2} b f + 17017 \, {\left (5 \, b^{3} c + 6 \, a b^{2} f\right )} x^{3} + 819 \, {\left (238 \, a b^{2} d + 27 \, a^{2} b g\right )} x + 8415 \, {\left (7 \, b^{3} x^{5} + 16 \, a b^{2} x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{765765 \, b^{2}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x}, x\right ) \]