Optimal. Leaf size=41 \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]
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Rubi [A] time = 0.0084468, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {78, 63, 206} \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\frac{11}{7 \sqrt{1-2 x}}-\frac{1}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{11}{7 \sqrt{1-2 x}}+\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0214941, size = 41, normalized size = 1. \[ \frac{11}{7 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 0.7 \begin{align*}{\frac{2\,\sqrt{21}}{147}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{11}{7}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.13395, size = 62, normalized size = 1.51 \begin{align*} -\frac{1}{147} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{11}{7 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6753, size = 149, normalized size = 3.63 \begin{align*} \frac{\sqrt{21}{\left (2 \, x - 1\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 231 \, \sqrt{-2 \, x + 1}}{147 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.0539, size = 78, normalized size = 1.9 \begin{align*} - \frac{2 \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 )}{7} + \frac{11}{7 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.5182, size = 66, normalized size = 1.61 \begin{align*} -\frac{1}{147} \, \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{11}{7 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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