Integrand size = 17, antiderivative size = 20 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \]
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Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (1-x \right )^{\frac {3}{2}}}\) | \(15\) |
default | \(\frac {2 \sqrt {1+x}}{3 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\) | \(30\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+2 x +1\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(49\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {x^{2} + {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.00 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\begin {cases} \frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx=\frac {\left (\frac {x\,\sqrt {x+1}}{3}+\frac {\sqrt {x+1}}{3}\right )\,\sqrt {1-x}}{x^2-2\,x+1} \]
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