Optimal. Leaf size=280 \[ -\frac{50299451003 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{4146187500 \sqrt{33}}+\frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]
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Rubi [A] time = 0.117976, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx &=\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{2}{75} \int \left (-\frac{71}{2}-\frac{89 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2} \left (-\frac{2339}{2}+\frac{2503 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{14625}\\ &=\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\sqrt{2+3 x} (3+5 x)^{5/2} \left (\frac{816405}{8}+\frac{599163 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{804375}\\ &=-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{113620371}{8}-\frac{172527627 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{36196875}\\ &=-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{(3+5 x)^{3/2} \left (\frac{22424965215}{16}+\frac{17105967765 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{760134375}\\ &=-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{735492282165}{8}-\frac{2263475295135 x}{16}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{11402015625}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\frac{95277493539765}{32}+\frac{37624103287695 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{102618140625}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{50299451003 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8292375000}+\frac{836091184171 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{22804031250}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.265368, size = 119, normalized size = 0.42 \[ \frac{\sqrt{2} \left (3344364736684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1684482853585 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+1316318850000 x^5+888419542500 x^4-227285730000 x^3-522917547750 x^2-177853891770 x+44426819351\right )}{273648375000} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 170, normalized size = 0.6 \begin{align*}{\frac{1}{8209451250000\,{x}^{3}+6293912625000\,{x}^{2}-1915538625000\,x-1641890250000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -492567075000000\,{x}^{9}-1562321722500000\,{x}^{8}-1592905277250000\,{x}^{7}-33511953825000\,{x}^{6}+1684482853585\,\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 ) -3344364736684\,\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 ) +1050958443600000\,{x}^{5}+633067124890500\,{x}^{4}-67989068522100\,{x}^{3}-162128981218890\,{x}^{2}-22684068454890\,x+7996827483180 \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{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{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 (-{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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