Optimal. Leaf size=27 \[ \frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{35} b x^{7/2} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ \frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{35} b x^{7/2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^{3/2} \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac {1}{5} (2 b) \int x^{5/2} \, dx\\ &=-\frac {4}{35} b x^{7/2}+\frac {2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 0.85 \[ \frac {2}{35} x^{5/2} \left (7 \tanh ^{-1}(\tanh (a+b x))-2 b x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 18, normalized size = 0.67 \[ \frac {2}{35} \, {\left (5 \, b x^{3} + 7 \, a x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 13, normalized size = 0.48 \[ \frac {2}{7} \, b x^{\frac {7}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 20, normalized size = 0.74 \[ -\frac {4 b \,x^{\frac {7}{2}}}{35}+\frac {2 x^{\frac {5}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 19, normalized size = 0.70 \[ -\frac {4}{35} \, b x^{\frac {7}{2}} + \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 57, normalized size = 2.11 \[ \frac {x^{5/2}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{5}-\frac {4\,b\,x^{7/2}}{35}-\frac {x^{5/2}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.46, size = 26, normalized size = 0.96 \[ - \frac {4 b x^{\frac {7}{2}}}{35} + \frac {2 x^{\frac {5}{2}} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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