Optimal. Leaf size=69 \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
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Rubi [A] time = 0.0220431, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {99, 156, 63, 206} \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x}}{3+5 x}+\int \frac{-5+3 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{\sqrt{1-2 x}}{3+5 x}+21 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-34 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{\sqrt{1-2 x}}{3+5 x}-21 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+34 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{\sqrt{1-2 x}}{3+5 x}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0428895, size = 69, normalized size = 1. \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 54, normalized size = 0.8 \begin{align*} -2\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+{\frac{2}{5}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{68\,\sqrt{55}}{55}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80736, size = 119, normalized size = 1.72 \begin{align*} -\frac{34}{55} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{\sqrt{-2 \, x + 1}}{5 \, x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63965, size = 251, normalized size = 3.64 \begin{align*} \frac{34 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, \sqrt{21}{\left (5 \, x + 3\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 55 \, \sqrt{-2 \, x + 1}}{55 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.3249, size = 230, normalized size = 3.33 \begin{align*} - 44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right ) + 42 \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 ) - 70 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68385, size = 127, normalized size = 1.84 \begin{align*} -\frac{34}{55} \, \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 ) + \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{\sqrt{-2 \, x + 1}}{5 \, x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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