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 [B] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {15, 435, 111, 110} \[ -\frac {2 \sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {-x} \sqrt {x} E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 110
Rule 111
Rule 435
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx &=\left (\sqrt {\frac {1}{x}} \sqrt {x}\right ) \int \frac {\sqrt {-1+\frac {1}{x}}}{\sqrt {1+x}} \, dx\\ &=\frac {\sqrt {-1+\frac {1}{x}} \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {\left (\sqrt {-1+\frac {1}{x}} \sqrt {-x}\right ) \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {1+x}} \, dx}{\sqrt {1-x} \sqrt {\frac {1}{x}} \sqrt {x}}\\ &=-\frac {2 \sqrt {-1+\frac {1}{x}} \sqrt {-x} E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}} \sqrt {x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 66, normalized size = 2.75 \[ -\frac {2 \sqrt {\frac {x}{x+1}} \sqrt {1-x^2} \left (x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )\right )}{3 \sqrt {1-x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-\frac {x - 1}{x}}}{\sqrt {x + 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 49, normalized size = 2.04 \[ -\frac {2 \sqrt {\frac {1}{x}}\, \sqrt {-\frac {x -1}{x}}\, \sqrt {-x}\, \sqrt {-2 x +2}\, \sqrt {x}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{x -1} \]
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 \[ \int \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {x}\,\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}}}{\sqrt {x+1}} \,d x \]
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 \[ \int \frac {\sqrt {x} \sqrt {-1 + \frac {1}{x}} \sqrt {\frac {1}{x}}}{\sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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