Optimal. Leaf size=72 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{4 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0316608, antiderivative size = 72, 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 = {2171, 2167} \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{4 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{(2 b) \int \frac{1}{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{4 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0405726, size = 46, normalized size = 0.64 \[ -\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (\tanh ^{-1}(\tanh (a+b x))-3 b x\right )}{3 x^{3/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.168, size = 59, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -3\,bx}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}^{-{\frac{3}{2}}}}+{\frac{4\,b}{3\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48287, size = 45, normalized size = 0.62 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{2} + a b x - a^{2}\right )}}{3 \, \sqrt{b x + a} a^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12291, size = 61, normalized size = 0.85 \begin{align*} \frac{2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15603, size = 74, normalized size = 1.03 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )} b^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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