Optimal. Leaf size=81 \[ -\frac {3}{5} \sqrt {1-x^2} x^2-\frac {3}{20} (5 x+8) \sqrt {1-x^2}-\frac {1}{5} \sqrt {1-x^2} x^4-\frac {1}{2} \sqrt {1-x^2} x^3+\frac {3}{4} \sin ^{-1}(x) \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1809, 833, 780, 216} \[ -\frac {1}{5} \sqrt {1-x^2} x^4-\frac {1}{2} \sqrt {1-x^2} x^3-\frac {3}{5} \sqrt {1-x^2} x^2-\frac {3}{20} (5 x+8) \sqrt {1-x^2}+\frac {3}{4} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 833
Rule 1809
Rubi steps
\begin {align*} \int \frac {x^3 (1+x)^2}{\sqrt {1-x^2}} \, dx &=-\frac {1}{5} x^4 \sqrt {1-x^2}-\frac {1}{5} \int \frac {(-9-10 x) x^3}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{2} x^3 \sqrt {1-x^2}-\frac {1}{5} x^4 \sqrt {1-x^2}+\frac {1}{20} \int \frac {x^2 (30+36 x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{5} x^2 \sqrt {1-x^2}-\frac {1}{2} x^3 \sqrt {1-x^2}-\frac {1}{5} x^4 \sqrt {1-x^2}-\frac {1}{60} \int \frac {(-72-90 x) x}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{5} x^2 \sqrt {1-x^2}-\frac {1}{2} x^3 \sqrt {1-x^2}-\frac {1}{5} x^4 \sqrt {1-x^2}-\frac {3}{20} (8+5 x) \sqrt {1-x^2}+\frac {3}{4} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{5} x^2 \sqrt {1-x^2}-\frac {1}{2} x^3 \sqrt {1-x^2}-\frac {1}{5} x^4 \sqrt {1-x^2}-\frac {3}{20} (8+5 x) \sqrt {1-x^2}+\frac {3}{4} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.52 \[ \frac {3}{4} \sin ^{-1}(x)-\frac {1}{20} \sqrt {1-x^2} \left (4 x^4+10 x^3+12 x^2+15 x+24\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 50, normalized size = 0.62 \[ -\frac {1}{20} \, {\left (4 \, x^{4} + 10 \, x^{3} + 12 \, x^{2} + 15 \, x + 24\right )} \sqrt {-x^{2} + 1} - \frac {3}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 34, normalized size = 0.42 \[ -\frac {1}{20} \, {\left ({\left (2 \, {\left ({\left (2 \, x + 5\right )} x + 6\right )} x + 15\right )} x + 24\right )} \sqrt {-x^{2} + 1} + \frac {3}{4} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.88 \[ -\frac {\sqrt {-x^{2}+1}\, x^{4}}{5}-\frac {\sqrt {-x^{2}+1}\, x^{3}}{2}-\frac {3 \sqrt {-x^{2}+1}\, x^{2}}{5}-\frac {3 \sqrt {-x^{2}+1}\, x}{4}+\frac {3 \arcsin \relax (x )}{4}-\frac {6 \sqrt {-x^{2}+1}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 70, normalized size = 0.86 \[ -\frac {1}{5} \, \sqrt {-x^{2} + 1} x^{4} - \frac {1}{2} \, \sqrt {-x^{2} + 1} x^{3} - \frac {3}{5} \, \sqrt {-x^{2} + 1} x^{2} - \frac {3}{4} \, \sqrt {-x^{2} + 1} x - \frac {6}{5} \, \sqrt {-x^{2} + 1} + \frac {3}{4} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 36, normalized size = 0.44 \[ \frac {3\,\mathrm {asin}\relax (x)}{4}-\sqrt {1-x^2}\,\left (\frac {x^4}{5}+\frac {x^3}{2}+\frac {3\,x^2}{5}+\frac {3\,x}{4}+\frac {6}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 73, normalized size = 0.90 \[ - \frac {x^{4} \sqrt {1 - x^{2}}}{5} - \frac {x^{3} \sqrt {1 - x^{2}}}{2} - \frac {3 x^{2} \sqrt {1 - x^{2}}}{5} - \frac {3 x \sqrt {1 - x^{2}}}{4} - \frac {6 \sqrt {1 - x^{2}}}{5} + \frac {3 \operatorname {asin}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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