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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 110
Rule 111
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*}
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Mathematica [C] time = 0.02, 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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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 1} \sqrt {x} \sqrt {-x + 1}}{x^{2} + x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {x + 1} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 1.04 \[ \frac {2 \sqrt {2}\, \sqrt {-x}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {x + 1} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {1-x}}{\sqrt {x}\,\sqrt {x+1}} \,d x \]
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 \[ \int \frac {\sqrt {1 - x}}{\sqrt {x} \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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