\[ \int \frac {x^3 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 x \left (e \,x^{2}+d x +c \right )}{3 b \sqrt {b \,x^{3}+a}}+\frac {4 e \sqrt {b \,x^{3}+a}}{3 b^{2}}+\frac {8 d \sqrt {b \,x^{3}+a}}{3 b^{\frac {5}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {4 a^{\frac {1}{3}} d \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {1}{4}}}{3 b^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {4 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (b^{\frac {1}{3}} c -2 a^{\frac {1}{3}} d \left (1-\sqrt {3}\right )\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{9 b^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^3*(e*x^2+d*x+c)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (b c x^{3} + a c\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 4 \, {\left (b d x^{3} + a d\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (b d x^{2} + b c x - {\left (b x^{3} + 2 \, a\right )} e\right )} \sqrt {b x^{3} + a}\right )}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e x^{5} + d x^{4} + c x^{3}\right )} \sqrt {b x^{3} + a}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]