Optimal. Leaf size=20 \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]
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 = 0.00, size = 20, normalized size = 1.00 \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.15, size = 33, normalized size = 1.65 \[ \frac {x^{2} + {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 19, normalized size = 0.95 \[ \frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.75 \[ \frac {\left (x +1\right )^{\frac {3}{2}}}{3 \left (-x +1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 38, normalized size = 1.90 \[ \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 34, normalized size = 1.70 \[ \frac {\left (\frac {x\,\sqrt {x+1}}{3}+\frac {\sqrt {x+1}}{3}\right )\,\sqrt {1-x}}{x^2-2\,x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 61, normalized size = 3.05 \[ \begin {cases} \frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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