Optimal. Leaf size=52 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A] time = 0.0258759, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2168, 2165} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2165
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=-\frac{2 \sqrt{x}}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{\int \frac{1}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}\\ \end{align*}
Mathematica [A] time = 0.0475296, size = 55, normalized size = 1.06 \[ \frac{2 \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 42, normalized size = 0.8 \begin{align*} -2\,{\frac{\sqrt{x}}{b\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}}+2\,{\frac{\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ) }{{b}^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10558, size = 308, normalized size = 5.92 \begin{align*} \left [\frac{{\left (b x + a\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \, \sqrt{b x + a} b \sqrt{x}}{b^{3} x + a b^{2}}, -\frac{2 \,{\left ({\left (b x + a\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) + \sqrt{b x + a} b \sqrt{x}\right )}}{b^{3} x + a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\operatorname{atanh}^{\frac{3}{2}}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22081, size = 53, normalized size = 1.02 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{b^{\frac{3}{2}}} - \frac{2 \, \sqrt{x}}{\sqrt{b x + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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