Optimal. Leaf size=125 \[ -\frac{202 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{539 \sqrt{33}}+\frac{544 \sqrt{3 x+2} \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]
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Rubi [A] time = 0.0422164, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{544 \sqrt{3 x+2} \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{231 (1-2 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{121}{2}-15 x}{(1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{231 (1-2 x)^{3/2}}+\frac{544 \sqrt{2+3 x} \sqrt{3+5 x}}{17787 \sqrt{1-2 x}}+\frac{4 \int \frac{\frac{885}{4}-1020 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{17787}\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{231 (1-2 x)^{3/2}}+\frac{544 \sqrt{2+3 x} \sqrt{3+5 x}}{17787 \sqrt{1-2 x}}-\frac{272 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5929}+\frac{101}{539} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{231 (1-2 x)^{3/2}}+\frac{544 \sqrt{2+3 x} \sqrt{3+5 x}}{17787 \sqrt{1-2 x}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.118468, size = 115, normalized size = 0.92 \[ -\frac{3605 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+4 \sqrt{3 x+2} \sqrt{5 x+3} (272 x-213)-272 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17787 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 228, normalized size = 1.8 \begin{align*} -{\frac{1}{17787\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 7210\,\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}-544\,\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}-3605\,\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 ) +272\,\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 ) +16320\,{x}^{3}+7892\,{x}^{2}-9660\,x-5112 \right ) \sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}} \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{1}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{120 \, x^{5} - 28 \, x^{4} - 90 \, x^{3} + 27 \, x^{2} + 17 \, x - 6}, 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{1}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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