Optimal. Leaf size=23 \[ \frac{2}{9} (x+1)^{3/2} \left (x^2-x+1\right )^{3/2} \]
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Rubi [A] time = 0.0178384, 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} \[ \frac{2}{9} (x+1)^{3/2} \left (x^2-x+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 913
Rubi steps
\begin{align*} \int x^2 \sqrt{1+x} \sqrt{1-x+x^2} \, dx &=\frac{2}{9} (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0344631, size = 23, normalized size = 1. \[ \frac{2}{9} (x+1)^{3/2} \left (x^2-x+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 18, normalized size = 0.8 \begin{align*}{\frac{2}{9} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ({x}^{2}-x+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47234, size = 30, normalized size = 1.3 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )} \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 = 2.19205, size = 61, normalized size = 2.65 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} \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 x^{2} \sqrt{x + 1} \sqrt{x^{2} - x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13955, size = 36, normalized size = 1.57 \begin{align*} \frac{2}{9} \, \sqrt{{\left (x + 1\right )}^{2} - 3 \, x}{\left ({\left (x + 1\right )}{\left (x - 2\right )} + 3\right )}{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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