Optimal. Leaf size=24 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0080433, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {111, 110} \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 111
Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x}}{\sqrt{x} \sqrt{1+x}} \, dx &=\frac{\sqrt{-x} \int \frac{\sqrt{1-x}}{\sqrt{-x} \sqrt{1+x}} \, dx}{\sqrt{x}}\\ &=-\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}}\\ \end{align*}
Mathematica [C] time = 0.0196932, size = 40, normalized size = 1.67 \[ -\frac{2}{3} \sqrt{x} \left (x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.009, size = 25, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) }{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{x + 1} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}{x^{2} + x}, 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{\sqrt{1 - x}}{\sqrt{x} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{x + 1} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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