Integrand size = 20, antiderivative size = 61 \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=-\frac {4 \sqrt {1+x}}{\sqrt {1-x}}-\sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}+3 \arcsin (x) \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {91, 21, 49, 52, 41, 222} \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=3 \arcsin (x)-\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}+\frac {(x+1)^{3/2}}{3 (1-x)^{3/2}}-3 \sqrt {1-x} \sqrt {x+1} \]
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Rule 21
Rule 41
Rule 49
Rule 52
Rule 91
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {1+x} (3+3 x)}{(1-x)^{3/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx \\ & = -3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -3 \sqrt {1-x} \sqrt {1+x}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}+3 \sin ^{-1}(x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=-\frac {\sqrt {1+x} \left (14-19 x+3 x^2\right )}{3 (1-x)^{3/2}}+6 \arctan \left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \]
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Time = 1.75 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.36
method | result | size |
default | \(\frac {\left (9 \arcsin \left (x \right ) x^{2}-3 x^{2} \sqrt {-x^{2}+1}-18 \arcsin \left (x \right ) x +19 x \sqrt {-x^{2}+1}+9 \arcsin \left (x \right )-14 \sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{3 \left (-1+x \right )^{2} \sqrt {-x^{2}+1}}\) | \(83\) |
risch | \(\frac {\left (3 x^{3}-16 x^{2}-5 x +14\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {3 \arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\) | \(84\) |
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Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=-\frac {14 \, x^{2} + {\left (3 \, x^{2} - 19 \, x + 14\right )} \sqrt {x + 1} \sqrt {-x + 1} + 18 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 28 \, x + 14}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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\[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=\int \frac {x^{2} \sqrt {x + 1}}{\left (1 - x\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=-\frac {{\left ({\left (3 \, x - 22\right )} {\left (x + 1\right )} + 36\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {x^2 \sqrt {1+x}}{(1-x)^{5/2}} \, dx=\int \frac {x^2\,\sqrt {x+1}}{{\left (1-x\right )}^{5/2}} \,d x \]
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