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 = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 77
Rule 6129
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.47, size = 24, normalized size = 0.67 \[ -\frac {2 \, {\left (x^{2} + 3 \, x + 6\right )} \sqrt {-x^{2} + 1}}{5 \, \sqrt {x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.78 \[ \frac {2 \left (-1+x \right ) \left (x^{2}+3 x +6\right ) \sqrt {1+x}}{5 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 22, normalized size = 0.61 \[ \frac {2 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x - 6\right )}}{5 \, \sqrt {-x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 45, normalized size = 1.25 \[ -\frac {2\,x^2\,\sqrt {1-x^2}+6\,x\,\sqrt {1-x^2}+12\,\sqrt {1-x^2}}{5\,\sqrt {x+1}} \]
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 \[ \int \frac {x \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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