\[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 c \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}+\frac {2 x \left (-b c \,x^{2}+a e x +a d \right )}{3 a^{2} \sqrt {b \,x^{3}+a}}+\frac {2 c \sqrt {b \,x^{3}+a}}{3 a^{2}}-\frac {2 e \sqrt {b \,x^{3}+a}}{3 a \,b^{\frac {2}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {e \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {1}{4}}}{3 a^{\frac {2}{3}} b^{\frac {2}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {2 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (b^{\frac {1}{3}} d +a^{\frac {1}{3}} e \left (1-\sqrt {3}\right )\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{9 a \,b^{\frac {2}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {4 \, {\left (a b x^{3} + a^{2}\right )} \sqrt {b} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (b^{2} c x^{3} + a b c\right )} \sqrt {a} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left (a b d x^{3} + a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 4 \, {\left (a b x^{2} e + a b d x + a b c\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {2 \, {\left (a b x^{3} + a^{2}\right )} \sqrt {b} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (b^{2} c x^{3} + a b c\right )} \sqrt {-a} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left (a b d x^{3} + a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (a b x^{2} e + a b d x + a b c\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a} {\left (e x^{2} + d x + c\right )}}{b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x}, x\right ) \]