\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{39 \left (2+3 x \right )^{\frac {13}{2}}}+\frac {230 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{1287 \left (2+3 x \right )^{\frac {11}{2}}}-\frac {75041008472 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1752079329}-\frac {2257166048 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1752079329}+\frac {1300 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{891 \left (2+3 x \right )^{\frac {9}{2}}}-\frac {3347620 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1702701 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {23210828 \sqrt {1-2 x}\, \sqrt {3+5 x}}{11918907 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {1079936248 \sqrt {1-2 x}\, \sqrt {3+5 x}}{83432349 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {75041008472 \sqrt {1-2 x}\, \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (27352447588044 \, x^{6} + 110328276131100 \, x^{5} + 185457331738206 \, x^{4} + 166295375376786 \, x^{3} + 83893544414217 \, x^{2} + 22577209892436 \, x + 2532151719515\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{584026443 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256}, x\right ) \]