\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {5594 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{124509}-\frac {1196 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{124509}+\frac {4 \sqrt {3+5 x}}{231 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x}\, \sqrt {2+3 x}}+\frac {5594 \sqrt {1-2 x}\, \sqrt {3+5 x}}{41503 \sqrt {2+3 x}} \]
command
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (33564 \, x^{2} - 39220 \, x + 12297\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{124509 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{360 \, x^{6} + 156 \, x^{5} - 326 \, x^{4} - 99 \, x^{3} + 105 \, x^{2} + 16 \, x - 12}, x\right ) \]