Optimal. Leaf size=251 \[ -\frac{201616}{49} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac{11171040 \sqrt{3 x+2} \sqrt{1-2 x}}{49 \sqrt{5 x+3}}-\frac{5544440 \sqrt{3 x+2} \sqrt{1-2 x}}{147 (5 x+3)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{2234208}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.102896, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac{11171040 \sqrt{3 x+2} \sqrt{1-2 x}}{49 \sqrt{5 x+3}}-\frac{5544440 \sqrt{3 x+2} \sqrt{1-2 x}}{147 (5 x+3)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{201616}{49} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2234208}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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)^{9/2} (3+5 x)^{5/2}} \, dx &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{2}{21} \int \frac{(231-231 x) \sqrt{1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{4}{315} \int \frac{-\frac{51975}{2}+39270 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{8 \int \frac{-\frac{5672205}{2}+\frac{7825125 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{6615}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16 \int \frac{-\frac{852857775}{4}+\frac{490002975 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{46305}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5544440 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{32 \int \frac{-\frac{34932969225}{4}+\frac{21612920175 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{1528065}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5544440 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{11171040 \sqrt{1-2 x} \sqrt{2+3 x}}{49 \sqrt{3+5 x}}-\frac{64 \int \frac{-\frac{909761408325}{8}-\frac{359255409975 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{16808715}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5544440 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{11171040 \sqrt{1-2 x} \sqrt{2+3 x}}{49 \sqrt{3+5 x}}+\frac{1108888}{49} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{6702624}{49} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac{44 \sqrt{1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{11924 \sqrt{1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2488904 \sqrt{1-2 x}}{441 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5544440 \sqrt{1-2 x} \sqrt{2+3 x}}{147 (3+5 x)^{3/2}}+\frac{11171040 \sqrt{1-2 x} \sqrt{2+3 x}}{49 \sqrt{3+5 x}}-\frac{2234208}{49} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{201616}{49} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.274498, size = 114, normalized size = 0.45 \[ \frac{2}{147} \left (12 \sqrt{2} \left (279276 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-140665 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (6786406800 x^5+21944379060 x^4+28367736228 x^3+18325125498 x^2+5915384456 x+763335749\right )}{(3 x+2)^{7/2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 501, normalized size = 2. \begin{align*}{\frac{2}{294\,x-147}\sqrt{1-2\,x} \left ( 227877300\,\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}-452427120\,\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}+592480980\,\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}-1176310512\,\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}+577289160\,\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}-1146148704\,\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}+249821040\,\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}-495994176\,\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}+13572813600\,{x}^{6}+40511520\,\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 ) -80431488\,\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 ) +37102351320\,{x}^{5}+34791093396\,{x}^{4}+8282514768\,{x}^{3}-6494356586\,{x}^{2}-4388712958\,x-763335749 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{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}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{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}}{30375 \, x^{8} + 155925 \, x^{7} + 350055 \, x^{6} + 448911 \, x^{5} + 359670 \, x^{4} + 184360 \, x^{3} + 59040 \, x^{2} + 10800 \, x + 864}, 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}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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