Optimal. Leaf size=218 \[ -\frac{1654421 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1913625 \sqrt{33}}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{9698 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{93555}-\frac{146963 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{467775}-\frac{1654421 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4209975}-\frac{146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3827250 \sqrt{33}} \]
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Rubi [A] time = 0.0806832, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2}{33} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{9698 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{93555}-\frac{146963 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{467775}-\frac{1654421 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4209975}-\frac{1654421 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1913625 \sqrt{33}}-\frac{146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3827250 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx &=\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{33} \int \frac{\left (-50-\frac{185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (-\frac{9245}{4}-\frac{24245 x}{4}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx}{4455}\\ &=\frac{9698 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{93555}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (\frac{84395}{4}-\frac{2204445 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{467775}\\ &=-\frac{146963 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{467775}+\frac{9698 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{93555}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{44328915}{16}+\frac{24816315 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{7016625}\\ &=-\frac{1654421 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4209975}-\frac{146963 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{467775}+\frac{9698 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{93555}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{-\frac{685297455}{8}-\frac{2193331695 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{63149625}\\ &=-\frac{1654421 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4209975}-\frac{146963 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{467775}+\frac{9698 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{93555}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{1654421 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3827250}+\frac{146222113 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{42099750}\\ &=-\frac{1654421 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{4209975}-\frac{146963 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{467775}+\frac{9698 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{93555}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2}{33} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3827250 \sqrt{33}}-\frac{1654421 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1913625 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.264092, size = 107, normalized size = 0.49 \[ \frac{-91626220 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4-12379500 x^3-16381350 x^2+9143865 x+3748468\right )+146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{63149625 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{3788977500\,{x}^{3}+2904882750\,{x}^{2}-884094750\,x-757795500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}+6463800000\,{x}^{6}+91626220\,\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 ) -146222113\,\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 ) -28643220000\,{x}^{5}-5066658000\,{x}^{4}+15351281550\,{x}^{3}+3614874270\,{x}^{2}-2433073980\,x-674724240 \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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 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{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, 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}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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