Optimal. Leaf size=222 \[ -\frac{48478 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3645}-\frac{6464 \sqrt{1-2 x} (5 x+3)^{5/2}}{81 \sqrt{3 x+2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{48478}{729} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{136028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645} \]
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Rubi [A] time = 0.0785786, 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 = {97, 150, 154, 158, 113, 119} \[ -\frac{6464 \sqrt{1-2 x} (5 x+3)^{5/2}}{81 \sqrt{3 x+2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{48478}{729} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{48478 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645}+\frac{136028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{2}{15} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{4}{135} \int \frac{\left (-295-\frac{4925 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{6464 \sqrt{1-2 x} (3+5 x)^{5/2}}{81 \sqrt{2+3 x}}+\frac{8}{405} \int \frac{\left (\frac{118045}{4}-\frac{206925 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{11036}{81} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{6464 \sqrt{1-2 x} (3+5 x)^{5/2}}{81 \sqrt{2+3 x}}-\frac{8 \int \frac{\left (\frac{137475}{2}-\frac{1817925 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{6075}\\ &=-\frac{48478}{729} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{6464 \sqrt{1-2 x} (3+5 x)^{5/2}}{81 \sqrt{2+3 x}}+\frac{8 \int \frac{-\frac{2121825}{8}-\frac{2550525 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{54675}\\ &=-\frac{48478}{729} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{6464 \sqrt{1-2 x} (3+5 x)^{5/2}}{81 \sqrt{2+3 x}}-\frac{136028 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3645}+\frac{266629 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3645}\\ &=-\frac{48478}{729} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{11036}{81} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac{6464 \sqrt{1-2 x} (3+5 x)^{5/2}}{81 \sqrt{2+3 x}}+\frac{136028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645}-\frac{48478 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3645}\\ \end{align*}
Mathematica [A] time = 0.244505, size = 109, normalized size = 0.49 \[ \frac{\sqrt{2} \left (935915 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-136028 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (24300 x^4-45090 x^3-461043 x^2-517257 x-158237\right )}{(3 x+2)^{5/2}}}{10935} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 324, normalized size = 1.5 \begin{align*} -{\frac{1}{109350\,{x}^{2}+10935\,x-32805} \left ( 8423235\,\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}-1224252\,\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}+11230980\,\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}-1632336\,\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}-1458000\,{x}^{6}+3743660\,\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 ) -544112\,\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 ) +2559600\,{x}^{5}+28370520\,{x}^{4}+32990058\,{x}^{3}+4298988\,{x}^{2}-8361204\,x-2848266 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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 (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, 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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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