Optimal. Leaf size=41 \[ \frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 41, 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 = {47, 37}
\begin {gather*} \frac {\sqrt {x+1}}{3 \sqrt {1-x}}+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {1}{3} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(-2+x) \sqrt {1+x}}{3 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 30, normalized size = 0.73
method | result | size |
gosper | \(-\frac {\sqrt {1+x}\, \left (-2+x \right )}{3 \left (1-x \right )^{\frac {3}{2}}}\) | \(18\) |
default | \(\frac {\sqrt {1+x}}{3 \left (1-x \right )^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\) | \(30\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}-x -2\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 38, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 39, normalized size = 0.95 \begin {gather*} \frac {2 \, x^{2} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - 4 \, x + 2}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.16, size = 128, normalized size = 3.12 \begin {gather*} \begin {cases} \frac {x + 1}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {3}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \left (x + 1\right )}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} + \frac {3 i}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 22, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 43, normalized size = 1.05 \begin {gather*} \frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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