Optimal. Leaf size=20 \[ -\sqrt {\frac {1-x}{x+1}} (x+1) \]
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Rubi [A] time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1962, 12, 383} \begin {gather*} -\sqrt {\frac {1-x}{x+1}} (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 383
Rule 1962
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {\frac {1-x}{1+x}} (1+x)} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1-x^2}{2 \left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-x}{1+x}}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-x}{1+x}}\right )\right )\\ &=-\sqrt {\frac {1-x}{1+x}} (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} \frac {x-1}{\sqrt {\frac {1-x}{x+1}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 1.60 \begin {gather*} -\frac {2 \sqrt {\frac {1-x}{x+1}}}{\frac {1-x}{x+1}+1} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 17, normalized size = 0.85 \begin {gather*} -{\left (x + 1\right )} \sqrt {-\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 29, normalized size = 1.45 \begin {gather*} -\frac {2}{\sqrt {-\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {-\frac {x - 1}{x + 1}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.85 \begin {gather*} \frac {x -1}{\sqrt {-\frac {x -1}{x +1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 27, normalized size = 1.35 \begin {gather*} \frac {2 \, \sqrt {-\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 17, normalized size = 0.85 \begin {gather*} -\sqrt {-\frac {x-1}{x+1}}\,\left (x+1\right ) \end {gather*}
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 \begin {gather*} \int \frac {x}{\sqrt {- \frac {x - 1}{x + 1}} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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