Integrand size = 20, antiderivative size = 63 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{24} (32+21 x) \sqrt {1-x^2}+\frac {7 \arcsin (x)}{8} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1823, 847, 794, 222} \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {7 \arcsin (x)}{8}-\frac {2}{3} \sqrt {1-x^2} x^2-\frac {1}{24} (21 x+32) \sqrt {1-x^2}-\frac {1}{4} \sqrt {1-x^2} x^3 \]
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Rule 222
Rule 794
Rule 847
Rule 1823
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{4} \int \frac {(-7-8 x) x^2}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}+\frac {1}{12} \int \frac {x (16+21 x)}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{24} (32+21 x) \sqrt {1-x^2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{24} (32+21 x) \sqrt {1-x^2}+\frac {7}{8} \sin ^{-1}(x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {1}{24} \sqrt {1-x^2} \left (-32-21 x-16 x^2-6 x^3\right )+\frac {7}{4} \arctan \left (\frac {x}{-1+\sqrt {1-x^2}}\right ) \]
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Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {\left (6 x^{3}+16 x^{2}+21 x +32\right ) \left (x^{2}-1\right )}{24 \sqrt {-x^{2}+1}}+\frac {7 \arcsin \left (x \right )}{8}\) | \(37\) |
trager | \(\left (-\frac {1}{4} x^{3}-\frac {2}{3} x^{2}-\frac {7}{8} x -\frac {4}{3}\right ) \sqrt {-x^{2}+1}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{8}\) | \(54\) |
default | \(-\frac {x^{3} \sqrt {-x^{2}+1}}{4}-\frac {7 x \sqrt {-x^{2}+1}}{8}+\frac {7 \arcsin \left (x \right )}{8}-\frac {2 x^{2} \sqrt {-x^{2}+1}}{3}-\frac {4 \sqrt {-x^{2}+1}}{3}\) | \(57\) |
meijerg | \(\frac {i \left (i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{2 \sqrt {\pi }}+\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 x^{2}+8\right ) \sqrt {-x^{2}+1}}{6}}{\sqrt {\pi }}-\frac {i \left (-\frac {i \sqrt {\pi }\, x \left (10 x^{2}+15\right ) \sqrt {-x^{2}+1}}{20}+\frac {3 i \sqrt {\pi }\, \arcsin \left (x \right )}{4}\right )}{2 \sqrt {\pi }}\) | \(102\) |
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{24} \, {\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt {-x^{2} + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=- \frac {x^{3} \sqrt {1 - x^{2}}}{4} - \frac {2 x^{2} \sqrt {1 - x^{2}}}{3} - \frac {7 x \sqrt {1 - x^{2}}}{8} - \frac {4 \sqrt {1 - x^{2}}}{3} + \frac {7 \operatorname {asin}{\left (x \right )}}{8} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \frac {2}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {7}{8} \, \sqrt {-x^{2} + 1} x - \frac {4}{3} \, \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.48 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x + 8\right )} x + 21\right )} x + 32\right )} \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.49 \[ \int \frac {x^2 (1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {7\,\mathrm {asin}\left (x\right )}{8}-\sqrt {1-x^2}\,\left (\frac {x^3}{4}+\frac {2\,x^2}{3}+\frac {7\,x}{8}+\frac {4}{3}\right ) \]
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