Optimal. Leaf size=28 \[ \frac{x^4}{4}-\frac{x^2}{\sqrt{2}}+\log \left (x^2+\sqrt{2}\right ) \]
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Rubi [A] time = 0.0191322, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{x^4}{4}-\frac{x^2}{\sqrt{2}}+\log \left (x^2+\sqrt{2}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{2}+x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2}+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\sqrt{2}+x+\frac{2}{\sqrt{2}+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{\sqrt{2}}+\frac{x^4}{4}+\log \left (\sqrt{2}+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0105252, size = 31, normalized size = 1.11 \[ \frac{1}{4} \left (x^4-2 \sqrt{2} x^2+4 \log \left (x^2+\sqrt{2}\right )-6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 23, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}+\ln \left ({x}^{2}+\sqrt{2} \right ) -{\frac{{x}^{2}\sqrt{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44337, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{2} \, \sqrt{2} x^{2} + \log \left (x^{2} + \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87988, size = 65, normalized size = 2.32 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{2} \, \sqrt{2} x^{2} + \log \left (x^{2} + \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114682, size = 24, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4} - \frac{\sqrt{2} x^{2}}{2} + \log{\left (x^{2} + \sqrt{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05794, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{2} \, \sqrt{2} x^{2} + \log \left (x^{2} + \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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