Optimal. Leaf size=87 \[ \frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {101, 156, 157,
12, 94, 212} \begin {gather*} \frac {14 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {5 \sqrt {x+1}}{3 \sqrt {1-x} x}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 101
Rule 156
Rule 157
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {2}{3} \int \frac {-\frac {5}{2}-2 x}{(1-x)^{3/2} x^2 \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+\frac {2}{3} \int \frac {\frac {9}{2}+\frac {5 x}{2}}{(1-x)^{3/2} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-\frac {2}{3} \int -\frac {9}{2 \sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+3 \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )\\ &=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 54, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {1+x} \left (3-19 x+14 x^2\right )}{3 (1-x)^{3/2} x}-6 \tanh ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 113, normalized size = 1.30
method | result | size |
risch | \(\frac {\left (14 x^{3}-5 x^{2}-16 x +3\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, x \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\) | \(95\) |
default | \(-\frac {\left (9 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{3}-18 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}+14 x^{2} \sqrt {-x^{2}+1}+9 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x -19 x \sqrt {-x^{2}+1}+3 \sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{3 x \left (-1+x \right )^{2} \sqrt {-x^{2}+1}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 86, normalized size = 0.99 \begin {gather*} \frac {14 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {3}{\sqrt {-x^{2} + 1}} + \frac {7 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {4}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - 3 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 84, normalized size = 0.97 \begin {gather*} \frac {13 \, x^{3} - 26 \, x^{2} - {\left (14 \, x^{2} - 19 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 9 \, {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 13 \, x}{3 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1}}{x^{2} \left (1 - x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x+1}}{x^2\,{\left (1-x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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