\[ \int \frac {(e x)^{7/2} \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx \]
Optimal antiderivative \[ -\frac {a \left (4 A b -3 a B \right ) e^{\frac {7}{2}} \arctanh \left (\frac {\left (e x \right )^{\frac {3}{2}} \sqrt {b}}{e^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}\right )}{12 b^{\frac {5}{2}}}+\frac {\left (4 A b -3 a B \right ) e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}{12 b^{2}}+\frac {B \left (e x \right )^{\frac {9}{2}} \sqrt {b \,x^{3}+a}}{6 b e} \]
command
integrate((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(1/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{24} \, {\left (B {\left (\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {5}{2}}} - \frac {2 \, {\left (\frac {5 \, \sqrt {b x^{3} + a} a^{2} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {9}{2}}}\right )}}{b^{4} - \frac {2 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}}}\right )} - 4 \, A {\left (\frac {a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {3}{2}}} - \frac {2 \, \sqrt {b x^{3} + a} a}{{\left (b^{2} - \frac {{\left (b x^{3} + a\right )} b}{x^{3}}\right )} x^{\frac {3}{2}}}\right )}\right )} e^{\frac {7}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{3} + A\right )} \left (e x\right )^{\frac {7}{2}}}{\sqrt {b x^{3} + a}}\,{d x} \]________________________________________________________________________________________