Optimal. Leaf size=36 \[ -\frac{2}{5} (1-x)^{5/2}+2 (1-x)^{3/2}-4 \sqrt{1-x} \]
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Rubi [A] time = 0.0415186, antiderivative size = 36, 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} \[ -\frac{2}{5} (1-x)^{5/2}+2 (1-x)^{3/2}-4 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 77
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} x \sqrt{1+x} \, dx &=\int \frac{x (1+x)}{\sqrt{1-x}} \, dx\\ &=\int \left (\frac{2}{\sqrt{1-x}}-3 \sqrt{1-x}+(1-x)^{3/2}\right ) \, dx\\ &=-4 \sqrt{1-x}+2 (1-x)^{3/2}-\frac{2}{5} (1-x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0083818, size = 21, normalized size = 0.58 \[ -\frac{2}{5} \sqrt{1-x} \left (x^2+3 x+6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 28, normalized size = 0.8 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ({x}^{2}+3\,x+6 \right ) }{5}\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 = 0.94335, size = 30, normalized size = 0.83 \begin{align*} \frac{2 \,{\left (x^{3} + 2 \, x^{2} + 3 \, x - 6\right )}}{5 \, \sqrt{-x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7774, size = 66, normalized size = 1.83 \begin{align*} -\frac{2 \,{\left (x^{2} + 3 \, x + 6\right )} \sqrt{-x^{2} + 1}}{5 \, \sqrt{x + 1}} \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{x \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28932, size = 51, normalized size = 1.42 \begin{align*} -\frac{2}{5} \,{\left (x - 1\right )}^{2} \sqrt{-x + 1} + 2 \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{8}{5} \, \sqrt{2} - 4 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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