Optimal. Leaf size=75 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2168, 2165} \[ -\frac {2 \sqrt {x}}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{5/2}}-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2165
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx &=-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {\int \frac {\sqrt {x}}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b^2}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{5/2}}-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 1.04 \[ \frac {2 \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x^{3/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 186, normalized size = 2.48 \[ \left [\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 49, normalized size = 0.65 \[ -\frac {2 \, \sqrt {x} {\left (\frac {4 \, x}{b} + \frac {3 \, a}{b^{2}}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}}} - \frac {2 \, \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 59, normalized size = 0.79 \[ -\frac {2 x^{\frac {3}{2}}}{3 b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}-\frac {2 \sqrt {x}}{b^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}+\frac {2 \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {3}{2}}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{3/2}}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {3}{2}}}{\operatorname {atanh}^{\frac {5}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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