Optimal. Leaf size=42 \[ \frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}} \]
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Rubi [A] time = 0.00, antiderivative size = 42, 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, 39} \[ \frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx &=\frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2}{3} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.71 \[ \frac {2 x^2-2 x-1}{3 (x-1) \sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 54, normalized size = 1.29 \[ \frac {x^{3} - x^{2} - {\left (2 \, x^{2} - 2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - x + 1}{3 \, {\left (x^{3} - x^{2} - x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 67, normalized size = 1.60 \[ \frac {\sqrt {2} - \sqrt {-x + 1}}{8 \, \sqrt {x + 1}} - \frac {{\left (5 \, x - 7\right )} \sqrt {x + 1} \sqrt {-x + 1}}{12 \, {\left (x - 1\right )}^{2}} - \frac {\sqrt {x + 1}}{8 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.60 \[ -\frac {2 x^{2}-2 x -1}{3 \sqrt {x +1}\, \left (-x +1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 40, normalized size = 0.95 \[ \frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} - \frac {1}{3 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 42, normalized size = 1.00 \[ \frac {2\,x\,\sqrt {1-x}+\sqrt {1-x}-2\,x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.28, size = 158, normalized size = 3.76 \[ \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 \sqrt {-1 + \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 i \sqrt {1 - \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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