Optimal. Leaf size=63 \[ -\frac{1}{4} \sqrt{1-x^2} x^3-\frac{2}{3} \sqrt{1-x^2} x^2-\frac{1}{24} (21 x+32) \sqrt{1-x^2}+\frac{7}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.080565, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1809, 833, 780, 216} \[ -\frac{1}{4} \sqrt{1-x^2} x^3-\frac{2}{3} \sqrt{1-x^2} x^2-\frac{1}{24} (21 x+32) \sqrt{1-x^2}+\frac{7}{8} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1809
Rule 833
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2 (1+x)^2}{\sqrt{1-x^2}} \, dx &=-\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*}
Mathematica [A] time = 0.0302614, size = 37, normalized size = 0.59 \[ \frac{7}{8} \sin ^{-1}(x)-\frac{1}{24} \sqrt{1-x^2} \left (6 x^3+16 x^2+21 x+32\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{4}\sqrt{-{x}^{2}+1}}-{\frac{7\,x}{8}\sqrt{-{x}^{2}+1}}+{\frac{7\,\arcsin \left ( x \right ) }{8}}-{\frac{2\,{x}^{2}}{3}\sqrt{-{x}^{2}+1}}-{\frac{4}{3}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48661, size = 76, normalized size = 1.21 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85394, size = 119, normalized size = 1.89 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.762138, size = 60, normalized size = 0.95 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11217, size = 41, normalized size = 0.65 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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