Optimal. Leaf size=58 \[ -\frac {2 \sqrt {1-x}}{3 \sqrt {x+1}}-\frac {2 \sqrt {1-x}}{3 (x+1)^{3/2}}+\frac {1}{(x+1)^{3/2} \sqrt {1-x}} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 \sqrt {1-x}}{3 \sqrt {x+1}}-\frac {2 \sqrt {1-x}}{3 (x+1)^{3/2}}+\frac {1}{(x+1)^{3/2} \sqrt {1-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}+2 \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx\\ &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}+\frac {2}{3} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.52 \begin {gather*} \frac {2 x^2+2 x-1}{3 \sqrt {1-x} (x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 48, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x+1} \left (-\frac {(1-x)^2}{(x+1)^2}-\frac {6 (1-x)}{x+1}+3\right )}{12 \sqrt {1-x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 49, normalized size = 0.84 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 108, normalized size = 1.86 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{96 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {7 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{32 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1} \sqrt {-x + 1}}{4 \, {\left (x - 1\right )}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {21 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{96 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.43 \begin {gather*} \frac {2 x^{2}+2 x -1}{3 \left (x +1\right )^{\frac {3}{2}} \sqrt {-x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 38, normalized size = 0.66 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} - \frac {1}{3 \, {\left (\sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 48, normalized size = 0.83 \begin {gather*} -\frac {2\,x\,\sqrt {1-x}-\sqrt {1-x}+2\,x^2\,\sqrt {1-x}}{\left (3\,x^2-3\right )\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.40, size = 165, normalized size = 2.84 \begin {gather*} \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {\sqrt {-1 + \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {i \sqrt {1 - \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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