Optimal. Leaf size=41 \[ -\frac {1}{3} \sqrt {1-x^2} x^2-\frac {1}{3} (3 x+5) \sqrt {1-x^2}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1809, 780, 216} \[ -\frac {1}{3} \sqrt {1-x^2} x^2-\frac {1}{3} (3 x+5) \sqrt {1-x^2}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 1809
Rubi steps
\begin {align*} \int \frac {x (1+x)^2}{\sqrt {1-x^2}} \, dx &=-\frac {1}{3} x^2 \sqrt {1-x^2}-\frac {1}{3} \int \frac {(-5-6 x) x}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{3} x^2 \sqrt {1-x^2}-\frac {1}{3} (5+3 x) \sqrt {1-x^2}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{3} x^2 \sqrt {1-x^2}-\frac {1}{3} (5+3 x) \sqrt {1-x^2}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.63 \[ \sin ^{-1}(x)-\frac {1}{3} \sqrt {1-x^2} \left (x^2+3 x+5\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 38, normalized size = 0.93 \[ -\frac {1}{3} \, {\left (x^{2} + 3 \, x + 5\right )} \sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 21, normalized size = 0.51 \[ -\frac {1}{3} \, {\left ({\left (x + 3\right )} x + 5\right )} \sqrt {-x^{2} + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 1.00 \[ -\frac {\sqrt {-x^{2}+1}\, x^{2}}{3}-\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )-\frac {5 \sqrt {-x^{2}+1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 40, normalized size = 0.98 \[ -\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \sqrt {-x^{2} + 1} x - \frac {5}{3} \, \sqrt {-x^{2} + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 22, normalized size = 0.54 \[ \mathrm {asin}\relax (x)-\sqrt {1-x^2}\,\left (\frac {x^2}{3}+x+\frac {5}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 37, normalized size = 0.90 \[ - \frac {x^{2} \sqrt {1 - x^{2}}}{3} - x \sqrt {1 - x^{2}} - \frac {5 \sqrt {1 - x^{2}}}{3} + \operatorname {asin}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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