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}{(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*}
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Mathematica [A]
time = 10.03, 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 = 25, normalized size = 1.09
method | result | size |
gosper | \(-\frac {2}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(18\) |
risch | \(-\frac {2}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(18\) |
default | \(-\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{3 \left (x^{3}+1\right )}\) | \(25\) |
elliptic | \(-\frac {2 \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \sqrt {x^{3}+1}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 2.13, size = 24, normalized size = 1.04 \begin {gather*} -\frac {2 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{3 \, {\left (x^{3} + 1\right )}} \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}}{\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.69, size = 17, normalized size = 0.74 \begin {gather*} -\frac {2}{3\,\sqrt {x+1}\,\sqrt {x^2-x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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