Optimal. Leaf size=25 \[ -\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2016, 446, 270}
\begin {gather*} -\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 446
Rule 2016
Rubi steps
\begin {align*} \int \frac {\left (\frac {2+x^2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx &=\int \frac {\left (1+\frac {2}{x^2}\right )^{7/9}}{\left (2+x^2\right )^{3/2}} \, dx\\ &=\frac {\left (\left (1+\frac {2}{x^2}\right )^{7/9} x^{14/9}\right ) \int \frac {1}{x^{14/9} \left (2+x^2\right )^{13/18}} \, dx}{\left (2+x^2\right )^{7/9}}\\ &=-\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}}\\ \end {align*}
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Mathematica [A]
time = 6.55, size = 25, normalized size = 1.00 \begin {gather*} -\frac {9 \left (1+\frac {2}{x^2}\right )^{7/9} x}{10 \sqrt {2+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 22, normalized size = 0.88
method | result | size |
gosper | \(-\frac {9 x \left (\frac {x^{2}+2}{x^{2}}\right )^{\frac {7}{9}}}{10 \sqrt {x^{2}+2}}\) | \(22\) |
risch | \(-\frac {9 x \left (\frac {x^{2}+2}{x^{2}}\right )^{\frac {7}{9}}}{10 \sqrt {x^{2}+2}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 21, normalized size = 0.84 \begin {gather*} -\frac {9 \, x \left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{10 \, \sqrt {x^{2} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.45, size = 15, normalized size = 0.60 \begin {gather*} -\frac {9\,x\,{\left (x^2+2\right )}^{5/18}\,{\left (\frac {1}{x^2}\right )}^{7/9}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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