\[ \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {a \left (10 A b -3 a B \right ) \left (e x \right )^{\frac {9}{2}} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{144 b e}+\frac {\left (10 A b -3 a B \right ) \left (e x \right )^{\frac {9}{2}} \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{120 b e}+\frac {B \left (e x \right )^{\frac {9}{2}} \left (b \,x^{3}+a \right )^{\frac {7}{2}}}{15 b e}-\frac {a^{4} \left (10 A b -3 a B \right ) e^{\frac {7}{2}} \arctanh \left (\frac {\left (e x \right )^{\frac {3}{2}} \sqrt {b}}{e^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}\right )}{384 b^{\frac {5}{2}}}+\frac {a^{3} \left (10 A b -3 a B \right ) e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}{384 b^{2}}+\frac {a^{2} \left (10 A b -3 a B \right ) \left (e x \right )^{\frac {9}{2}} \sqrt {b \,x^{3}+a}}{192 b e} \]
command
integrate((e*x)^(7/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{11520} \, {\left (10 \, {\left (\frac {15 \, a^{4} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left (\frac {15 \, \sqrt {b x^{3} + a} a^{4} b^{3}}{x^{\frac {3}{2}}} - \frac {55 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {9}{2}}} + \frac {73 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {15}{2}}} + \frac {15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {21}{2}}}\right )}}{b^{5} - \frac {4 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {6 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {4 \, {\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac {{\left (b x^{3} + a\right )}^{4} b}{x^{12}}}\right )} A - 3 \, {\left (\frac {15 \, a^{5} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (\frac {15 \, \sqrt {b x^{3} + a} a^{5} b^{4}}{x^{\frac {3}{2}}} - \frac {70 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{5} b^{3}}{x^{\frac {9}{2}}} + \frac {128 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{5} b^{2}}{x^{\frac {15}{2}}} + \frac {70 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{5} b}{x^{\frac {21}{2}}} - \frac {15 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} a^{5}}{x^{\frac {27}{2}}}\right )}}{b^{7} - \frac {5 \, {\left (b x^{3} + a\right )} b^{6}}{x^{3}} + \frac {10 \, {\left (b x^{3} + a\right )}^{2} b^{5}}{x^{6}} - \frac {10 \, {\left (b x^{3} + a\right )}^{3} b^{4}}{x^{9}} + \frac {5 \, {\left (b x^{3} + a\right )}^{4} b^{3}}{x^{12}} - \frac {{\left (b x^{3} + a\right )}^{5} b^{2}}{x^{15}}}\right )} B\right )} e^{\frac {7}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}}\,{d x} \]________________________________________________________________________________________