Optimal. Leaf size=18 \[ -\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ -\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx &=-\frac {(1+x)^{3/2}}{3 (-1+x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 31, normalized size = 1.72 \[ -\frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {x - 1} + x^{2} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 12, normalized size = 0.67 \[ -\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x - 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.72 \[ -\frac {\left (x +1\right )^{\frac {3}{2}}}{3 \left (x -1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 34, normalized size = 1.89 \[ -\frac {2 \, \sqrt {x^{2} - 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {x^{2} - 1}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 27, normalized size = 1.50 \[ -\frac {x\,\sqrt {x+1}+\sqrt {x+1}}{\left (3\,x-3\right )\,\sqrt {x-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 63, normalized size = 3.50 \[ \begin {cases} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {i \left (x + 1\right )^{\frac {3}{2}}}{- 3 \sqrt {1 - x} \left (x + 1\right ) + 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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