Optimal. Leaf size=26 \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]
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Rubi [A] time = 0.0080107, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {54, 215} \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]
Antiderivative was successfully verified.
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Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+2 x} \sqrt{2+3 x}} \, dx &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{13+3 x^2}} \, dx,x,\sqrt{-3+2 x}\right )\\ &=\sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{-3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0083718, size = 26, normalized size = 1. \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 57, normalized size = 2.2 \begin{align*}{\frac{\sqrt{6}}{6}\sqrt{ \left ( -3+2\,x \right ) \left ( 2+3\,x \right ) }\ln \left ({\frac{\sqrt{6}}{6} \left ( -{\frac{5}{2}}+6\,x \right ) }+\sqrt{6\,{x}^{2}-5\,x-6} \right ){\frac{1}{\sqrt{-3+2\,x}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44174, size = 38, normalized size = 1.46 \begin{align*} \frac{1}{6} \, \sqrt{6} \log \left (2 \, \sqrt{6} \sqrt{6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98652, size = 146, normalized size = 5.62 \begin{align*} \frac{1}{12} \, \sqrt{3} \sqrt{2} \log \left (4 \, \sqrt{3} \sqrt{2}{\left (12 \, x - 5\right )} \sqrt{3 \, x + 2} \sqrt{2 \, x - 3} + 288 \, x^{2} - 240 \, x - 119\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0987, size = 58, normalized size = 2.23 \begin{align*} \begin{cases} \frac{\sqrt{6} \operatorname{acosh}{\left (\frac{\sqrt{78} \sqrt{x + \frac{2}{3}}}{13} \right )}}{3} & \text{for}\: \frac{6 \left |{x + \frac{2}{3}}\right |}{13} > 1 \\- \frac{\sqrt{6} i \operatorname{asin}{\left (\frac{\sqrt{78} \sqrt{x + \frac{2}{3}}}{13} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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