7.275 Problem number 2907

\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx \]

Optimal antiderivative \[ -\frac {189368 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1764735}-\frac {23012 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1764735}+\frac {11 \sqrt {3+5 x}}{7 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}-\frac {229 \sqrt {1-2 x}\, \sqrt {3+5 x}}{343 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {2818 \sqrt {1-2 x}\, \sqrt {3+5 x}}{12005 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {5438 \sqrt {1-2 x}\, \sqrt {3+5 x}}{84035 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {189368 \sqrt {1-2 x}\, \sqrt {3+5 x}}{588245 \sqrt {2+3 x}} \]

command

integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {2 \, {\left (5112936 \, x^{4} + 7326810 \, x^{3} + 1004571 \, x^{2} - 2279324 \, x - 809083\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{588245 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32}, x\right ) \]