Optimal. Leaf size=222 \[ -\frac{1986944 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{27783}+\frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{66055016 \sqrt{5 x+3} \sqrt{1-2 x}}{27783 \sqrt{3 x+2}}+\frac{950584 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 (3 x+2)^{3/2}}+\frac{20420 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{512 \sqrt{5 x+3} \sqrt{1-2 x}}{81 (3 x+2)^{7/2}}-\frac{66055016 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783} \]
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Rubi [A] time = 0.0803941, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{66055016 \sqrt{5 x+3} \sqrt{1-2 x}}{27783 \sqrt{3 x+2}}+\frac{950584 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 (3 x+2)^{3/2}}+\frac{20420 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{512 \sqrt{5 x+3} \sqrt{1-2 x}}{81 (3 x+2)^{7/2}}-\frac{1986944 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783}-\frac{66055016 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt{3+5 x}} \, dx &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{(194-157 x) \sqrt{1-2 x}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}-\frac{4}{567} \int \frac{-\frac{31157}{2}+21301 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}+\frac{20420 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}-\frac{8 \int \frac{-\frac{2372055}{2}+\frac{2680125 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{19845}\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}+\frac{20420 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{950584 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{3/2}}-\frac{16 \int \frac{-\frac{205814595}{4}+\frac{62382075 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{416745}\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}+\frac{20420 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{950584 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{3/2}}+\frac{66055016 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 \sqrt{2+3 x}}-\frac{32 \int \frac{-\frac{2744348775}{4}-\frac{4334860425 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2917215}\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}+\frac{20420 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{950584 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{3/2}}+\frac{66055016 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 \sqrt{2+3 x}}+\frac{10928192 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{27783}+\frac{66055016 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{27783}\\ &=\frac{14 (1-2 x)^{3/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{512 \sqrt{1-2 x} \sqrt{3+5 x}}{81 (2+3 x)^{7/2}}+\frac{20420 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{950584 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 (2+3 x)^{3/2}}+\frac{66055016 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 \sqrt{2+3 x}}-\frac{66055016 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783}-\frac{1986944 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783}\\ \end{align*}
Mathematica [A] time = 0.266561, size = 111, normalized size = 0.5 \[ \frac{8 \left (\sqrt{2} \left (8256877 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4158805 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2675228148 x^4+7223771916 x^3+7318104714 x^2+3296666850 x+557240459\right )}{4 (3 x+2)^{9/2}}\right )}{83349} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 504, normalized size = 2.3 \begin{align*}{\frac{2}{833490\,{x}^{2}+83349\,x-250047} \left ( 1347452820\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-2675228148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+3593207520\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7133941728\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3593207520\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7133941728\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1596981120\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3170640768\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+80256844440\,{x}^{6}+266163520\,\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 ) -528440128\,\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 ) +224738841924\,{x}^{5}+217137403836\,{x}^{4}+55840372398\,{x}^{3}-39255728106\,{x}^{2}-27998280273\,x-5015164131 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{11}{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 (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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