Optimal. Leaf size=41 \[ -\frac {2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ -\frac {2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 47
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}-\int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.20 \[ \frac {-8 x^2+4 x+4}{3 \sqrt {1-x} (x+1)^{3/2}}-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 71, normalized size = 1.73 \[ \frac {2 \, {\left (2 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 4 \, x + 2\right )}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 102, normalized size = 2.49 \[ \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{12 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{4 \, \sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {15 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{12 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 73, normalized size = 1.78 \[ \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}-\frac {4 \left (2 x^{2}-x -1\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{3 \left (x +1\right )^{\frac {3}{2}} \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.01, size = 66, normalized size = 1.61 \[ -\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (1-x\right )}^{3/2}}{{\left (x+1\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.28, size = 126, normalized size = 3.07 \[ \begin {cases} \frac {8 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {4 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} + i \log {\left (x + 1 \right )} + 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {4 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} - 2 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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