\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {105584 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{135}+\frac {3176 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{135}+\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{15 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x}\, \sqrt {3+5 x}}-\frac {105584 \sqrt {1-2 x}\, \sqrt {2+3 x}}{27 \sqrt {3+5 x}} \]
command
integrate((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2375640 \, x^{3} + 4672674 \, x^{2} + 3061396 \, x + 668031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{45 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144}, x\right ) \]