Optimal. Leaf size=41 \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]
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Rubi [A] time = 0.0095627, 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} \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\frac{7}{11 \sqrt{1-2 x}}+\frac{1}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7}{11 \sqrt{1-2 x}}-\frac{1}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0191361, size = 41, normalized size = 1. \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 0.7 \begin{align*} -{\frac{2\,\sqrt{55}}{605}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{7}{11}{\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 = 1.50222, size = 62, normalized size = 1.51 \begin{align*} \frac{1}{605} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7}{11 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55368, size = 149, normalized size = 3.63 \begin{align*} \frac{\sqrt{55}{\left (2 \, x - 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 385 \, \sqrt{-2 \, x + 1}}{605 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.029, size = 78, normalized size = 1.9 \begin{align*} \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{11} + \frac{7}{11 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.5036, size = 66, normalized size = 1.61 \begin{align*} \frac{1}{605} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7}{11 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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