Optimal. Leaf size=191 \[ -\frac{62092 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{14175}-\frac{1}{9} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{62092 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2835}-\frac{8256877 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{56700} \]
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Rubi [A] time = 0.0669651, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{1}{9} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{62092 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2835}-\frac{62092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14175}-\frac{8256877 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{56700} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{9} \int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{173}{2}+135 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{3}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{315} \int \frac{\left (-\frac{18455}{2}-\frac{28155 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{1877}{630} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{2421135}{4}+931380 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{4725}\\ &=-\frac{62092 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2835}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{\int \frac{-\frac{78409965}{4}-\frac{123853155 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{42525}\\ &=-\frac{62092 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2835}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{341506 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{14175}+\frac{8256877 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{56700}\\ &=-\frac{62092 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2835}-\frac{1877}{630} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{3}{7} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{8256877 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{56700}-\frac{62092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14175}\\ \end{align*}
Mathematica [A] time = 0.225426, size = 105, normalized size = 0.55 \[ \frac{8256877 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (831761 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (47250 x^3+148950 x^2+212175 x+208073\right )\right )}{85050 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{5103000\,{x}^{3}+3912300\,{x}^{2}-1190700\,x-1020600}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -42525000\,{x}^{6}+4158805\,\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 ) -8256877\,\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 ) -166657500\,{x}^{5}-283810500\,{x}^{4}-293881950\,{x}^{3}-72202620\,{x}^{2}+81886830\,x+37453140 \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 (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}\,{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 (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{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 \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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