\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (3003 g \,x^{5}+3465 f \,x^{4}+4095 e \,x^{3}+5005 d \,x^{2}+6435 c x \right )}{45045 x^{2}}-\frac {2 a^{\frac {3}{2}} d \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {2 a^{2} g \sqrt {b \,x^{3}+a}}{15 b}-\frac {27 a c \sqrt {b \,x^{3}+a}}{7 x}+\frac {2 a \left (1001 g \,x^{5}+1485 f \,x^{4}+2457 e \,x^{3}+5005 d \,x^{2}+19305 c x \right ) \sqrt {b \,x^{3}+a}}{15015 x^{2}}+\frac {27 a \left (2 a f +13 b c \right ) \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 {4}{3}} \left (2 a f +13 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}}}}{182 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 {9 \,3^{\frac {3}{4}} a^{\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 (182 a^{\frac {2}{3}} b^{\frac {1}{3}} e -55 \left (2 a f +13 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}}}}{5005 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}}}} \]
command
integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {15015 \, a^{\frac {3}{2}} b d x \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 88452 \, a^{2} \sqrt {b} x e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 26730 \, {\left (13 \, a b c + 2 \, a^{2} f\right )} \sqrt {b} x {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 2 \, {\left (6006 \, b^{2} g x^{7} + 6930 \, b^{2} f x^{6} + 2002 \, {\left (5 \, b^{2} d + 6 \, a b g\right )} x^{4} + 990 \, {\left (13 \, b^{2} c + 16 \, a b f\right )} x^{3} - 45045 \, a b c + 2002 \, {\left (20 \, a b d + 3 \, a^{2} g\right )} x + 1638 \, {\left (5 \, b^{2} x^{5} + 14 \, a b x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{90090 \, b x}, \frac {15015 \, \sqrt {-a} a b d x \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) + 44226 \, a^{2} \sqrt {b} x e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 13365 \, {\left (13 \, a b c + 2 \, a^{2} f\right )} \sqrt {b} x {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (6006 \, b^{2} g x^{7} + 6930 \, b^{2} f x^{6} + 2002 \, {\left (5 \, b^{2} d + 6 \, a b g\right )} x^{4} + 990 \, {\left (13 \, b^{2} c + 16 \, a b f\right )} x^{3} - 45045 \, a b c + 2002 \, {\left (20 \, a b d + 3 \, a^{2} g\right )} x + 1638 \, {\left (5 \, b^{2} x^{5} + 14 \, a b x^{2}\right )} e\right )} \sqrt {b x^{3} + a}}{45045 \, b x}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{2}}, x\right ) \]