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 = 0.0569161, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 1991
Rule 435
Rule 264
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*}
Mathematica [A] time = 0.0085718, size = 25, normalized size = 1. \[ -\frac{9 \left (\frac{2}{x^2}+1\right )^{7/9} x}{10 \sqrt{x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 22, normalized size = 0.9 \begin{align*} -{\frac{9\,x}{10} \left ({\frac{{x}^{2}+2}{{x}^{2}}} \right ) ^{{\frac{7}{9}}}{\frac{1}{\sqrt{{x}^{2}+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x^{2} + 2}{x^{2}}\right )^{\frac{7}{9}}}{{\left (x^{2} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.55282, size = 61, normalized size = 2.44 \begin{align*} -\frac{9 \, x \left (\frac{x^{2} + 2}{x^{2}}\right )^{\frac{7}{9}}}{10 \, \sqrt{x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{x^{2} + 2}{x^{2}}\right )^{\frac{7}{9}}}{{\left (x^{2} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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