Optimal. Leaf size=35 \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]
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Rubi [A] time = 0.0203036, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {700, 1130, 206, 204} \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]
Antiderivative was successfully verified.
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Rule 700
Rule 1130
Rule 206
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x}}{1-x^2} \, dx &=6 \operatorname{Subst}\left (\int \frac{x^2}{5+4 x^2-x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=5 \operatorname{Subst}\left (\int \frac{1}{5-x^2} \, dx,x,\sqrt{2+3 x}\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{2+3 x}\right )\\ &=-\tan ^{-1}\left (\sqrt{2+3 x}\right )+\sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{2+3 x}}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.0151169, size = 35, normalized size = 1. \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 29, normalized size = 0.8 \begin{align*} -\arctan \left ( \sqrt{2+3\,x} \right ) +{\it Artanh} \left ({\frac{\sqrt{5}}{5}\sqrt{2+3\,x}} \right ) \sqrt{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65505, size = 61, normalized size = 1.74 \begin{align*} -\frac{1}{2} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{3 \, x + 2}}{\sqrt{5} + \sqrt{3 \, x + 2}}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32939, size = 116, normalized size = 3.31 \begin{align*} \frac{1}{2} \, \sqrt{5} \log \left (\frac{2 \, \sqrt{5} \sqrt{3 \, x + 2} + 3 \, x + 7}{x - 1}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.09664, size = 70, normalized size = 2. \begin{align*} - 5 \left (\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left (\frac{\sqrt{5} \sqrt{3 x + 2}}{5} \right )}}{5} & \text{for}\: 3 x + 2 > 5 \\- \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{3 x + 2}}{5} \right )}}{5} & \text{for}\: 3 x + 2 < 5 \end{cases}\right ) - \operatorname{atan}{\left (\sqrt{3 x + 2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32168, size = 65, normalized size = 1.86 \begin{align*} -\frac{1}{2} \, \sqrt{5} \log \left (\frac{{\left | -2 \, \sqrt{5} + 2 \, \sqrt{3 \, x + 2} \right |}}{2 \,{\left (\sqrt{5} + \sqrt{3 \, x + 2}\right )}}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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