\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{3 \left (2+3 x \right )^{\frac {7}{2}} \left (3+5 x \right )^{\frac {3}{2}}}-\frac {201616 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{147}-\frac {2234208 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{49}+\frac {44 \sqrt {1-2 x}}{3 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {11924 \sqrt {1-2 x}}{63 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {2488904 \sqrt {1-2 x}}{441 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {5544440 \sqrt {1-2 x}\, \sqrt {2+3 x}}{147 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {11171040 \sqrt {1-2 x}\, \sqrt {2+3 x}}{49 \sqrt {3+5 x}} \]
command
integrate((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (6786406800 \, x^{5} + 21944379060 \, x^{4} + 28367736228 \, x^{3} + 18325125498 \, x^{2} + 5915384456 \, x + 763335749\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{147 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{30375 \, x^{8} + 155925 \, x^{7} + 350055 \, x^{6} + 448911 \, x^{5} + 359670 \, x^{4} + 184360 \, x^{3} + 59040 \, x^{2} + 10800 \, x + 864}, x\right ) \]