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 = 0.05, antiderivative size = 49, 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} \[ \frac {2}{7} (1-x)^{7/2}-2 (1-x)^{5/2}+\frac {16}{3} (1-x)^{3/2}-8 \sqrt {1-x} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6129
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.62, size = 31, normalized size = 0.63 \[ -\frac {2 \, {\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right )} \sqrt {-x^{2} + 1}}{21 \, \sqrt {x + 1}} \]
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 = 35, normalized size = 0.71 \[ \frac {2 \left (-1+x \right ) \left (3 x^{3}+12 x^{2}+23 x +46\right ) \sqrt {1+x}}{21 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 29, normalized size = 0.59 \[ \frac {2 \, {\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt {-x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 52, normalized size = 1.06 \[ -\frac {\sqrt {1-x^2}\,\left (\frac {46\,x\,\sqrt {x+1}}{21}+\frac {92\,\sqrt {x+1}}{21}+\frac {8\,x^2\,\sqrt {x+1}}{7}+\frac {2\,x^3\,\sqrt {x+1}}{7}\right )}{x+1} \]
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 \[ \int \frac {x \left (x + 1\right )^{\frac {5}{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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