Optimal. Leaf size=25 \[ -\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \]
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Rubi [A] time = 0.06, 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 = {1991, 435, 264} \[ -\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 435
Rule 1991
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 = 0.01, size = 25, normalized size = 1.00 \[ -\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 8.54, size = 25, normalized size = 1.00 \[ -\frac {9 \left (\frac {2}{x^2}+1\right )^{7/9} x}{10 \sqrt {x^2+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 21, normalized size = 0.84 \[ -\frac {9 \, x \left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{10 \, \sqrt {x^{2} + 2}} \]
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 \[ \int \frac {\left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{{\left (x^{2} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, 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 \[ \int \frac {\left (\frac {x^{2} + 2}{x^{2}}\right )^{\frac {7}{9}}}{{\left (x^{2} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\frac {9\,x\,{\left (x^2+2\right )}^{5/18}\,{\left (\frac {1}{x^2}\right )}^{7/9}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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