Optimal. Leaf size=49 \[ \frac{2}{7} (1-x)^{7/2}-2 (1-x)^{5/2}+\frac{16}{3} (1-x)^{3/2}-8 \sqrt{1-x} \]
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Rubi [A] time = 0.0479878, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {6129, 77} \[ \frac{2}{7} (1-x)^{7/2}-2 (1-x)^{5/2}+\frac{16}{3} (1-x)^{3/2}-8 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 77
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} x (1+x)^{3/2} \, dx &=\int \frac{x (1+x)^2}{\sqrt{1-x}} \, dx\\ &=\int \left (\frac{4}{\sqrt{1-x}}-8 \sqrt{1-x}+5 (1-x)^{3/2}-(1-x)^{5/2}\right ) \, dx\\ &=-8 \sqrt{1-x}+\frac{16}{3} (1-x)^{3/2}-2 (1-x)^{5/2}+\frac{2}{7} (1-x)^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0111833, size = 28, normalized size = 0.57 \[ -\frac{2}{21} \sqrt{1-x} \left (3 x^3+12 x^2+23 x+46\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 35, normalized size = 0.7 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 3\,{x}^{3}+12\,{x}^{2}+23\,x+46 \right ) }{21}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94956, size = 39, normalized size = 0.8 \begin{align*} \frac{2 \,{\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt{-x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80191, size = 85, normalized size = 1.73 \begin{align*} -\frac{2 \,{\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right )} \sqrt{-x^{2} + 1}}{21 \, \sqrt{x + 1}} \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 = 1.28964, size = 70, normalized size = 1.43 \begin{align*} -\frac{2}{7} \,{\left (x - 1\right )}^{3} \sqrt{-x + 1} - 2 \,{\left (x - 1\right )}^{2} \sqrt{-x + 1} + \frac{16}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{64}{21} \, \sqrt{2} - 8 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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