Optimal. Leaf size=98 \[ -\frac{202}{225} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{4}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{272}{225} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0294153, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {102, 158, 113, 119} \[ -\frac{4}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{202}{225} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{272}{225} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 102
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=-\frac{4}{45} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{45} \int \frac{\frac{59}{2}-136 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{4}{45} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{272}{225} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx+\frac{1111}{225} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{4}{45} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{272}{225} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{202}{225} \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 = 0.147019, size = 92, normalized size = 0.94 \[ \frac{1}{675} \left (3605 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-272 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 140, normalized size = 1.4 \begin{align*} -{\frac{1}{20250\,{x}^{3}+15525\,{x}^{2}-4725\,x-4050}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 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 ) +1800\,{x}^{3}+1380\,{x}^{2}-420\,x-360 \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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3} \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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{15 \, x^{2} + 19 \, x + 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (1 - 2 x\right )^{\frac{3}{2}}}{\sqrt{3 x + 2} \sqrt{5 x + 3}}\, dx \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{3}{2}}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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