Optimal. Leaf size=38 \[ -\frac{2}{5} (1-x)^{5/2}+\frac{8}{3} (1-x)^{3/2}-8 \sqrt{1-x} \]
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Rubi [A] time = 0.0288797, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 43} \[ -\frac{2}{5} (1-x)^{5/2}+\frac{8}{3} (1-x)^{3/2}-8 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} (1+x)^{3/2} \, dx &=\int \frac{(1+x)^2}{\sqrt{1-x}} \, dx\\ &=\int \left (\frac{4}{\sqrt{1-x}}-4 \sqrt{1-x}+(1-x)^{3/2}\right ) \, dx\\ &=-8 \sqrt{1-x}+\frac{8}{3} (1-x)^{3/2}-\frac{2}{5} (1-x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0089038, size = 23, normalized size = 0.61 \[ -\frac{2}{15} \sqrt{1-x} \left (3 x^2+14 x+43\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 30, normalized size = 0.8 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 3\,{x}^{2}+14\,x+43 \right ) }{15}\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 = 0.953294, size = 32, normalized size = 0.84 \begin{align*} \frac{2 \,{\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt{-x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67183, size = 73, normalized size = 1.92 \begin{align*} -\frac{2 \,{\left (3 \, x^{2} + 14 \, x + 43\right )} \sqrt{-x^{2} + 1}}{15 \, \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 = 1.1949, size = 51, normalized size = 1.34 \begin{align*} -\frac{2}{5} \,{\left (x - 1\right )}^{2} \sqrt{-x + 1} + \frac{8}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{64}{15} \, \sqrt{2} - 8 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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