\[ \int \frac {(e x)^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -B a \right ) \left (e x \right )^{\frac {5}{2}}}{9 a b e \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 \left (4 A b +5 B a \right ) \left (e x \right )^{\frac {5}{2}}}{27 a^{2} b e \sqrt {b \,x^{3}+a}}-\frac {2 \left (4 A b +5 B a \right ) e \left (1+\sqrt {3}\right ) \sqrt {e x}\, \sqrt {b \,x^{3}+a}}{27 a^{2} b^{\frac {5}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )}+\frac {2 \left (4 A b +5 B a \right ) e \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}}}{27 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {5}{3}} b^{\frac {5}{3}} \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 +5 B a \right ) e \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}}}{81 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {5}{3}} b^{\frac {5}{3}} \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((e*x)^(3/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left ({\left (5 \, B a b^{2} + 4 \, A b^{3}\right )} x^{7} + 2 \, {\left (5 \, B a^{2} b + 4 \, A a b^{2}\right )} x^{4} + {\left (5 \, B a^{3} + 4 \, A a^{2} b\right )} x\right )} \sqrt {a} e^{\frac {3}{2}} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + {\left (5 \, B a^{3} + 4 \, A a^{2} b + {\left (8 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{27 \, {\left (a^{2} b^{4} x^{7} + 2 \, a^{3} b^{3} x^{4} + a^{4} b^{2} x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e x^{4} + A e x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]