Optimal. Leaf size=112 \[ \frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}-\frac {11}{2} \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {7 \sqrt {x+1}}{6 \sqrt {1-x} x^2}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}+\frac {26 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {19 \sqrt {x+1}}{6 \sqrt {1-x} x}-\frac {11}{2} \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^3} \, dx &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {2}{3} \int \frac {-\frac {7}{2}-3 x}{(1-x)^{3/2} x^3 \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}+\frac {1}{3} \int \frac {\frac {19}{2}+7 x}{(1-x)^{3/2} x^2 \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}-\frac {1}{3} \int \frac {-\frac {33}{2}-\frac {19 x}{2}}{(1-x)^{3/2} x \sqrt {1+x}} \, dx\\ &=\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}+\frac {1}{3} \int \frac {33}{2 \sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}+\frac {11}{2} \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}-\frac {11}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )\\ &=\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}-\frac {11}{2} \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 59, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {1+x} \left (3+12 x-71 x^2+52 x^3\right )}{6 (1-x)^{3/2} x^2}-11 \tanh ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 129, normalized size = 1.15
method | result | size |
risch | \(\frac {\left (52 x^{4}-19 x^{3}-59 x^{2}+15 x +3\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 x^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \left (-1+x \right ) \sqrt {1-x}\, \sqrt {1+x}}-\frac {11 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(100\) |
default | \(-\frac {\left (33 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{4}-66 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{3}+52 x^{3} \sqrt {-x^{2}+1}+33 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}-71 x^{2} \sqrt {-x^{2}+1}+12 x \sqrt {-x^{2}+1}+3 \sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{6 x^{2} \left (-1+x \right )^{2} \sqrt {-x^{2}+1}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 100, normalized size = 0.89 \begin {gather*} \frac {26 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {11}{2 \, \sqrt {-x^{2} + 1}} + \frac {13 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {11}{6 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - \frac {1}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} - \frac {11}{2} \, \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 = 0.52, size = 95, normalized size = 0.85 \begin {gather*} \frac {38 \, x^{4} - 76 \, x^{3} + 38 \, x^{2} - {\left (52 \, x^{3} - 71 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 33 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right )}{6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\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^{3} \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^3\,{\left (1-x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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