\[ \int \frac {A+B x^3}{\sqrt {e x} \left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -B a \right ) \sqrt {e x}}{9 a b e \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 \left (8 A b +B a \right ) \sqrt {e x}}{27 a^{2} b e \sqrt {b \,x^{3}+a}}+\frac {2 \left (8 A b +B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right ) \EllipticF \left (\sqrt {1-\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}, \frac {\sqrt {6}}{4}+\frac {\sqrt {2}}{4}\right ) \sqrt {e x}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{81 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {7}{3}} b e \sqrt {b \,x^{3}+a}\, \sqrt {\frac {b^{\frac {1}{3}} x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)/(b*x^3+a)^(5/2)/(e*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, {\left ({\left (B a b^{2} + 8 \, A b^{3}\right )} x^{6} + B a^{3} + 8 \, A a^{2} b + 2 \, {\left (B a^{2} b + 8 \, A a b^{2}\right )} x^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right ) + {\left (2 \, B a^{3} - 11 \, A a^{2} b - {\left (B a^{2} b + 8 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{27 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{b^{3} e x^{10} + 3 \, a b^{2} e x^{7} + 3 \, a^{2} b e x^{4} + a^{3} e x}, x\right ) \]