Integrand size = 17, antiderivative size = 87 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35 \arcsin (x)}{2} \]
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Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 52, 41, 222} \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {35 \arcsin (x)}{2}-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {x+1}}+\frac {35}{6} \sqrt {x+1} (1-x)^{3/2}+\frac {35}{2} \sqrt {x+1} \sqrt {1-x} \]
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Rule 41
Rule 49
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}-\frac {7}{3} \int \frac {(1-x)^{5/2}}{(1+x)^{3/2}} \, dx \\ & = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{3} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx \\ & = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx \\ & = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {\sqrt {1-x} \left (164+229 x+30 x^2-3 x^3\right )}{6 (1+x)^{3/2}}-35 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\left (3 x^{4}-33 x^{3}-199 x^{2}+65 x +164\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(84\) |
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Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {164 \, x^{2} - {\left (3 \, x^{3} - 30 \, x^{2} - 229 \, x - 164\right )} \sqrt {x + 1} \sqrt {-x + 1} - 210 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 328 \, x + 164}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 16.83 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.44 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\begin {cases} - 35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} + \frac {15 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {41 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {176 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {32 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {15 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {41 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {176}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {32}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {6 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{6} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {61 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {44 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {16 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {82}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, \arcsin \left (x\right ) \]
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Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {1}{2} \, \sqrt {x + 1} {\left (x - 12\right )} \sqrt {-x + 1} + \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {13 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {39 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 35 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\int \frac {{\left (1-x\right )}^{7/2}}{{\left (x+1\right )}^{5/2}} \,d x \]
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