Optimal. Leaf size=68 \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]
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Rubi [A] time = 0.0142218, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 47, 63, 206} \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^3} \, dx &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}+\frac{23}{14} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}-\frac{23}{42} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}+\frac{23}{42} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0388847, size = 69, normalized size = 1.01 \[ \frac{21 \left (142 x^2+19 x-45\right )+46 \sqrt{21-42 x} (3 x+2)^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{882 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 0.7 \begin{align*} -36\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{756}}+{\frac{23\,\sqrt{1-2\,x}}{108}} \right ) }+{\frac{23\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5965, size = 100, normalized size = 1.47 \begin{align*} -\frac{23}{882} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{21 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36016, size = 192, normalized size = 2.82 \begin{align*} \frac{23 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (71 \, x + 45\right )} \sqrt{-2 \, x + 1}}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 118.096, size = 316, normalized size = 4.65 \begin{align*} - \frac{148 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{20 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.05992, size = 92, normalized size = 1.35 \begin{align*} -\frac{23}{882} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{84 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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