Integrand size = 20, antiderivative size = 88 \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7 \arcsin (x)}{8} \]
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Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {92, 81, 52, 41, 222} \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\frac {7 \arcsin (x)}{8}-\frac {1}{4} \sqrt {1-x} x (x+1)^{5/2}-\frac {1}{6} \sqrt {1-x} (x+1)^{5/2}-\frac {7}{24} \sqrt {1-x} (x+1)^{3/2}-\frac {7}{8} \sqrt {1-x} \sqrt {x+1} \]
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Rule 41
Rule 52
Rule 81
Rule 92
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}-\frac {1}{4} \int \frac {(-1-2 x) (1+x)^{3/2}}{\sqrt {1-x}} \, dx \\ & = -\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{12} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx \\ & = -\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx \\ & = -\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \sin ^{-1}(x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {\sqrt {1-x} \left (32+53 x+37 x^2+22 x^3+6 x^4\right )}{24 \sqrt {1+x}}-\frac {7}{4} \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \]
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Time = 0.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (-6 x^{3} \sqrt {-x^{2}+1}-16 x^{2} \sqrt {-x^{2}+1}-21 x \sqrt {-x^{2}+1}+21 \arcsin \left (x \right )-32 \sqrt {-x^{2}+1}\right )}{24 \sqrt {-x^{2}+1}}\) | \(80\) |
risch | \(\frac {\left (6 x^{3}+16 x^{2}+21 x +32\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {7 \arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8 \sqrt {1-x}\, \sqrt {1+x}}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.59 \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{24} \, {\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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\[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\int \frac {x^{2} \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \frac {2}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {7}{8} \, \sqrt {-x^{2} + 1} x - \frac {4}{3} \, \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=-\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )} + 7\right )} {\left (x + 1\right )} + 21\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx=\int \frac {x^2\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \]
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