Optimal. Leaf size=23 \[ -\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]
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Rubi [A] time = 0.0211387, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {913} \[ -\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]
Antiderivative was successfully verified.
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Rule 913
Rubi steps
\begin{align*} \int \frac{x^2}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx &=-\frac{2}{3 \sqrt{1+x} \sqrt{1-x+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0304251, size = 23, normalized size = 1. \[ -\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 18, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{{x}^{2}-x+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47613, size = 23, normalized size = 1. \begin{align*} -\frac{2}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74493, size = 62, normalized size = 2.7 \begin{align*} -\frac{2 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}}{3 \,{\left (x^{3} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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