Optimal. Leaf size=191 \[ -\frac{509189 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8859375}+\frac{2}{45} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}}{1575}+\frac{8878 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}+\frac{21547 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1771875}-\frac{8024546 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375} \]
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Rubi [A] time = 0.0648209, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{45} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}}{1575}+\frac{8878 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}+\frac{21547 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1771875}-\frac{509189 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375}-\frac{8024546 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx &=\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{2}{45} \int \frac{\left (-\frac{113}{2}-\frac{159 x}{2}\right ) (1-2 x)^{3/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}{1575}+\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{4 \int \frac{\left (-\frac{5853}{2}-\frac{13317 x}{4}\right ) \sqrt{1-2 x} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx}{4725}\\ &=\frac{8878 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}{1575}+\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{8 \int \frac{\sqrt{2+3 x} \left (-\frac{545025}{8}+\frac{64641 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{354375}\\ &=\frac{21547 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1771875}+\frac{8878 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}{1575}+\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{8 \int \frac{\frac{32249013}{16}+\frac{12036819 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5315625}\\ &=\frac{21547 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1771875}+\frac{8878 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}{1575}+\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{5601079 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{17718750}+\frac{8024546 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{8859375}\\ &=\frac{21547 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1771875}+\frac{8878 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}{1575}+\frac{2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{8024546 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375}-\frac{509189 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375}\\ \end{align*}
Mathematica [A] time = 0.236693, size = 102, normalized size = 0.53 \[ \frac{754145 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1030500 x^2-113490 x+683887\right )+16049092 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{26578125 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 155, normalized size = 0.8 \begin{align*} -{\frac{1}{1594687500\,{x}^{3}+1222593750\,{x}^{2}-372093750\,x-318937500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -850500000\,{x}^{6}+754145\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +16049092\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +275400000\,{x}^{5}+1011636000\,{x}^{4}-583495200\,{x}^{3}-681204930\,{x}^{2}+123188070\,x+123099660 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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