Optimal. Leaf size=35 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0141844, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2167} \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x^{5/2}} \, dx &=\frac{2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0311113, size = 34, normalized size = 0.97 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{3/2} \left (3 b x-3 \tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.138, size = 29, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -3\,bx} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48269, size = 20, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99663, size = 46, normalized size = 1.31 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \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{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14085, size = 80, normalized size = 2.29 \begin{align*} \frac{4 \,{\left (3 \, b^{\frac{3}{2}}{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{4} + a^{2} b^{\frac{3}{2}}\right )}}{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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