\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {17888580643 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{6237000}+\frac {269045681 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3118500}+\frac {\left (2+3 x \right )^{\frac {7}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{\sqrt {1-2 x}}+\frac {419 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{66}+\frac {18 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{11}+\frac {4066493 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{23100}+\frac {9741 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{385}+\frac {269045681 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{207900} \]
command
integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (12757500 \, x^{5} + 60196500 \, x^{4} + 133330950 \, x^{3} + 198895770 \, x^{2} + 273928969 \, x - 477155552\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{207900 \, {\left (2 \, x - 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]