Integrand size = 17, antiderivative size = 81 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}} \]
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Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {2 \sqrt {x+1}}{35 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{35 (1-x)^{3/2}}+\frac {3 \sqrt {x+1}}{35 (1-x)^{5/2}}+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3}{7} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx \\ & = \frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {6}{35} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx \\ & = \frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2}{35} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx \\ & = \frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x} \left (12-13 x+8 x^2-2 x^3\right )}{35 (1-x)^{7/2}} \]
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Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(-\frac {\sqrt {1+x}\, \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \left (1-x \right )^{\frac {7}{2}}}\) | \(30\) |
default | \(\frac {\sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}+\frac {3 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{4}-6 x^{3}+5 x^{2}+x -12\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(59\) |
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Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {12 \, x^{4} - 48 \, x^{3} + 72 \, x^{2} - {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x - 12\right )} \sqrt {x + 1} \sqrt {-x + 1} - 48 \, x + 12}{35 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 19.32 (sec) , antiderivative size = 542, normalized size of antiderivative = 6.69 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\begin {cases} \frac {2 \left (x + 1\right )^{3}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {14 \left (x + 1\right )^{2}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} + \frac {35 \left (x + 1\right )}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {35}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \left (x + 1\right )^{3}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {14 i \left (x + 1\right )^{2}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} - \frac {35 i \left (x + 1\right )}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {35 i}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x - 1\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=-\frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )} - 35\right )} \sqrt {x + 1} \sqrt {-x + 1}}{35 \, {\left (x - 1\right )}^{4}} \]
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Time = 0.56 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=-\frac {x\,\sqrt {1-x}-12\,\sqrt {1-x}+5\,x^2\,\sqrt {1-x}-6\,x^3\,\sqrt {1-x}+2\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \]
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