Optimal. Leaf size=23 \[ \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {927}
\begin {gather*} \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 927
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx &=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}\\ \end {align*}
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Mathematica [A]
time = 10.05, size = 23, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 18, normalized size = 0.78
method | result | size |
gosper | \(\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3}\) | \(18\) |
default | \(\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3}\) | \(18\) |
risch | \(\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3}\) | \(18\) |
elliptic | \(\frac {2 \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \sqrt {x^{3}+1}}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (x^{3} + 1\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.23, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, \sqrt {{\left (x + 1\right )}^{2} - 3 \, x} \sqrt {x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 9, normalized size = 0.39 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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