Optimal. Leaf size=38 \[ 12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]
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Rubi [A] time = 0.0304311, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {826, 1166, 207} \[ 12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]
Antiderivative was successfully verified.
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Rule 826
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{13-x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=34 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )-36 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=12 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )}{\sqrt{15}}\\ \end{align*}
Mathematica [A] time = 0.0139014, size = 38, normalized size = 1. \[ 12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 44, normalized size = 1.2 \begin{align*} -{\frac{34\,\sqrt{15}}{15}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) -6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46203, size = 82, normalized size = 2.16 \begin{align*} \frac{17}{15} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5823, size = 163, normalized size = 4.29 \begin{align*} \frac{17}{15} \, \sqrt{15} \log \left (-\frac{\sqrt{15} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.7741, size = 95, normalized size = 2.5 \begin{align*} 34 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} > \frac{3}{5} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} < \frac{3}{5} \end{cases}\right ) - 6 \log{\left (-1 + \frac{1}{\sqrt{2 x + 3}} \right )} + 6 \log{\left (1 + \frac{1}{\sqrt{2 x + 3}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10009, size = 88, normalized size = 2.32 \begin{align*} \frac{17}{15} \, \sqrt{15} \log \left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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