Integrand size = 18, antiderivative size = 61 \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\arcsin (x) \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {81, 52, 41, 222} \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\arcsin (x)-\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{3} \sqrt {1-x} (x+1)^{3/2}-\sqrt {1-x} \sqrt {x+1} \]
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Rule 41
Rule 52
Rule 81
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {2}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx \\ & = -\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx \\ & = -\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\sin ^{-1}(x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {\sqrt {1-x} \left (5+8 x+4 x^2+x^3\right )}{3 \sqrt {1+x}}-2 \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \]
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Time = 1.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (-x^{2} \sqrt {-x^{2}+1}-3 x \sqrt {-x^{2}+1}+3 \arcsin \left (x \right )-5 \sqrt {-x^{2}+1}\right )}{3 \sqrt {-x^{2}+1}}\) | \(66\) |
risch | \(\frac {\left (x^{2}+3 x +5\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {\arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\) | \(74\) |
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Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{3} \, {\left (x^{2} + 3 \, x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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\[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\int \frac {x \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66 \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \sqrt {-x^{2} + 1} x - \frac {5}{3} \, \sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.61 \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{3} \, {\left ({\left (x + 2\right )} {\left (x + 1\right )} + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\int \frac {x\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \]
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