\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4}{231 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}-\frac {412810345784 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3691069305}-\frac {12417792656 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3691069305}+\frac {632}{5929 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {3606 \sqrt {1-2 x}}{207515 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {649224 \sqrt {1-2 x}}{1452605 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {6208896328 \sqrt {1-2 x}\, \sqrt {2+3 x}}{67110351 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {412810345784 \sqrt {1-2 x}\, \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}} \]
command
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (557293966808400 \, x^{6} + 873229924799280 \, x^{5} + 84649478011164 \, x^{4} - 430611138612568 \, x^{3} - 149619576926754 \, x^{2} + 52875828155808 \, x + 23506658680609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3691069305 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81000 \, x^{10} + 240300 \, x^{9} + 210330 \, x^{8} - 41619 \, x^{7} - 160643 \, x^{6} - 58821 \, x^{5} + 28917 \, x^{4} + 22192 \, x^{3} + 936 \, x^{2} - 2160 \, x - 432}, x\right ) \]