\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {184636 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{871563}-\frac {9124 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{871563}+\frac {4 \sqrt {3+5 x}}{231 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}}}+\frac {1072 \sqrt {3+5 x}}{17787 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}+\frac {974 \sqrt {1-2 x}\, \sqrt {3+5 x}}{41503 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {184636 \sqrt {1-2 x}\, \sqrt {3+5 x}}{290521 \sqrt {2+3 x}} \]
command
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3323448 \, x^{3} - 1066908 \, x^{2} - 1478206 \, x + 597945\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{871563 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1080 \, x^{7} + 1188 \, x^{6} - 666 \, x^{5} - 949 \, x^{4} + 117 \, x^{3} + 258 \, x^{2} - 4 \, x - 24}, x\right ) \]