Integrand size = 17, antiderivative size = 62 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {1}{5 (1-x)^{5/2} \sqrt {1+x}}+\frac {1}{5 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{5 \sqrt {1-x} \sqrt {1+x}} \]
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Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39} \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {2 x}{5 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{5 (1-x)^{3/2} \sqrt {x+1}}+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}} \]
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Rule 39
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5 (1-x)^{5/2} \sqrt {1+x}}+\frac {3}{5} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{5 (1-x)^{5/2} \sqrt {1+x}}+\frac {1}{5 (1-x)^{3/2} \sqrt {1+x}}+\frac {2}{5} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{5 (1-x)^{5/2} \sqrt {1+x}}+\frac {1}{5 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{5 \sqrt {1-x} \sqrt {1+x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {2+x-4 x^2+2 x^3}{5 (1-x)^{5/2} \sqrt {1+x}} \]
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Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(\frac {2 x^{3}-4 x^{2}+x +2}{5 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}\) | \(28\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{3}-4 x^{2}+x +2\right )}{5 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(54\) |
default | \(\frac {1}{5 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {1}{5 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {2}{5 \sqrt {1-x}\, \sqrt {1+x}}-\frac {2 \sqrt {1-x}}{5 \sqrt {1+x}}\) | \(58\) |
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none
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {2 \, x^{4} - 4 \, x^{3} - {\left (2 \, x^{3} - 4 \, x^{2} + x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x - 2}{5 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x - 1\right )}} \]
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Result contains complex when optimal does not.
Time = 10.31 (sec) , antiderivative size = 284, normalized size of antiderivative = 4.58 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {10 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac {15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {5 \sqrt {-1 + \frac {2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {10 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac {15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {5 i \sqrt {1 - \frac {2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {2 \, x}{5 \, \sqrt {-x^{2} + 1}} + \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{16 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{16 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left (11 \, x - 39\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{40 \, {\left (x - 1\right )}^{3}} \]
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Time = 0.38 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx=-\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-4\,x^2\,\sqrt {1-x}+2\,x^3\,\sqrt {1-x}}{5\,{\left (x-1\right )}^3\,\sqrt {x+1}} \]
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