3.354 \(\int \frac{x^5}{\sqrt{2}+x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^4}{4}-\frac{x^2}{\sqrt{2}}+\log \left (x^2+\sqrt{2}\right ) \]

[Out]

-(x^2/Sqrt[2]) + x^4/4 + Log[Sqrt[2] + x^2]

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Rubi [A]  time = 0.0455685, 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 \[ \frac{x^4}{4}-\frac{x^2}{\sqrt{2}}+\log \left (x^2+\sqrt{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(Sqrt[2] + x^2),x]

[Out]

-(x^2/Sqrt[2]) + x^4/4 + Log[Sqrt[2] + x^2]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\log{\left (x^{2} + \sqrt{2} \right )}}{4} - \frac{\int ^{x^{2}} \sqrt{2}\, dx}{2} + \frac{\int ^{x^{2}} x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**2+2**(1/2)),x)

[Out]

log(x**2 + sqrt(2))/4 - Integral(sqrt(2), (x, x**2))/2 + Integral(x, (x, x**2))/
2

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Mathematica [A]  time = 0.0143151, 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.

[In]  Integrate[x^5/(Sqrt[2] + x^2),x]

[Out]

(-6 - 2*Sqrt[2]*x^2 + x^4 + 4*Log[Sqrt[2] + x^2])/4

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Maple [A]  time = 0.01, size = 23, normalized size = 0.8 \[{\frac{{x}^{4}}{4}}+\ln \left ({x}^{2}+\sqrt{2} \right ) -{\frac{{x}^{2}\sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^2+2^(1/2)),x)

[Out]

1/4*x^4+ln(x^2+2^(1/2))-1/2*x^2*2^(1/2)

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Maxima [A]  time = 1.48534, size = 30, normalized size = 1.07 \[ \frac{1}{4} \, x^{4} - \frac{1}{2} \, \sqrt{2} x^{2} + \log \left (x^{2} + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 + sqrt(2)),x, algorithm="maxima")

[Out]

1/4*x^4 - 1/2*sqrt(2)*x^2 + log(x^2 + sqrt(2))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 + sqrt(2)),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.143149, size = 24, normalized size = 0.86 \[ \frac{x^{4}}{4} - \frac{\sqrt{2} x^{2}}{2} + \log{\left (x^{2} + \sqrt{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**2+2**(1/2)),x)

[Out]

x**4/4 - sqrt(2)*x**2/2 + log(x**2 + sqrt(2))

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GIAC/XCAS [A]  time = 0.205903, size = 30, normalized size = 1.07 \[ \frac{1}{4} \, x^{4} - \frac{1}{2} \, \sqrt{2} x^{2} +{\rm ln}\left (x^{2} + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 + sqrt(2)),x, algorithm="giac")

[Out]

1/4*x^4 - 1/2*sqrt(2)*x^2 + ln(x^2 + sqrt(2))