Integrand size = 20, antiderivative size = 112 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x^2}-\frac {7 \sqrt {1+x}}{6 \sqrt {1-x} x^2}-\frac {19 \sqrt {1+x}}{6 \sqrt {1-x} x}-\frac {11}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {101, 156, 157, 12, 94, 212} \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=-\frac {11}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {7 \sqrt {x+1}}{6 \sqrt {1-x} x^2}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}+\frac {26 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {19 \sqrt {x+1}}{6 \sqrt {1-x} x} \]
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Rule 12
Rule 94
Rule 101
Rule 156
Rule 157
Rule 212
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=-\frac {\sqrt {1+x} \left (3+12 x-71 x^2+52 x^3\right )}{6 (1-x)^{3/2} x^2}-11 \text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \]
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Time = 0.61 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {\left (52 x^{4}-19 x^{3}-59 x^{2}+15 x +3\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 x^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \left (-1+x \right ) \sqrt {1-x}\, \sqrt {1+x}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(100\) |
default | \(-\frac {\left (33 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{4}-66 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{3}+52 x^{3} \sqrt {-x^{2}+1}+33 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}-71 x^{2} \sqrt {-x^{2}+1}+12 x \sqrt {-x^{2}+1}+3 \sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{6 x^{2} \left (-1+x \right )^{2} \sqrt {-x^{2}+1}}\) | \(129\) |
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {38 \, x^{4} - 76 \, x^{3} + 38 \, x^{2} - {\left (52 \, x^{3} - 71 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 33 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right )}{6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )}} \]
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\[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\int \frac {\sqrt {x + 1}}{x^{3} \left (1 - x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {26 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {11}{2 \, \sqrt {-x^{2} + 1}} + \frac {13 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {11}{6 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - \frac {1}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} - \frac {11}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (82) = 164\).
Time = 0.35 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=-\frac {{\left (17 \, x - 19\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} - \frac {2 \, {\left (5 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} - \frac {28 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {28 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{2}} - \frac {11}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {11}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\int \frac {\sqrt {x+1}}{x^3\,{\left (1-x\right )}^{5/2}} \,d x \]
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