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