\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {16564 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{405}-\frac {496 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{405}+\frac {14 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{15 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {1736 \sqrt {1-2 x}\, \sqrt {3+5 x}}{135 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {16564 \sqrt {1-2 x}\, \sqrt {3+5 x}}{135 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (74538 \, x^{2} + 101862 \, x + 34927\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{135 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48}, x\right ) \]