Optimal. Leaf size=41 \[ \frac {\sqrt {x+1}}{3 \sqrt {1-x}}+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 37} \begin {gather*} \frac {\sqrt {x+1}}{3 \sqrt {1-x}}+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {1}{3} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(x-2) \sqrt {x+1}}{3 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 33, normalized size = 0.80 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {x+1}{1-x}+3\right )}{6 \sqrt {1-x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 39, normalized size = 0.95 \begin {gather*} \frac {2 \, x^{2} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - 4 \, x + 2}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 22, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {x +1}\, \left (x -2\right )}{3 \left (-x +1\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.12, size = 38, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 43, normalized size = 1.05 \begin {gather*} \frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.25, size = 139, normalized size = 3.39 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )}{3 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 i \sqrt {-1 + \frac {2}{x + 1}}} - \frac {3 i}{3 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 i \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {x + 1}{- 3 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 6 i \sqrt {1 - \frac {2}{x + 1}}} + \frac {3}{- 3 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 6 i \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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