Optimal. Leaf size=43 \[ -\sqrt{1-x} \sqrt{x+1}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.0113992, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {102, 157, 41, 216, 92, 206} \[ -\sqrt{1-x} \sqrt{x+1}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 102
Rule 157
Rule 41
Rule 216
Rule 92
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^{3/2}}{\sqrt{1-x} x} \, dx &=-\sqrt{1-x} \sqrt{1+x}-\int \frac{-1-2 x}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}+2 \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx+\int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}+2 \int \frac{1}{\sqrt{1-x^2}} \, dx-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=-\sqrt{1-x} \sqrt{1+x}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.041401, size = 64, normalized size = 1.49 \[ \frac{x^2+\sqrt{1-x^2} \sin ^{-1}(x)-1}{\sqrt{1-x^2}}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 51, normalized size = 1.2 \begin{align*}{\sqrt{1-x}\sqrt{1+x} \left ( -\sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) +2\,\arcsin \left ( x \right ) \right ){\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 = 1.74653, size = 55, normalized size = 1.28 \begin{align*} -\sqrt{-x^{2} + 1} + 2 \, \arcsin \left (x\right ) - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02555, size = 151, normalized size = 3.51 \begin{align*} -\sqrt{x + 1} \sqrt{-x + 1} - 4 \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \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{\left (x + 1\right )^{\frac{3}{2}}}{x \sqrt{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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