3.4.54 \(\int \frac {x^5}{\sqrt {2}+x^2} \, dx\) [354]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \frac {x^4}{4}-\frac {x^2}{\sqrt {2}}+\log \left (x^2+\sqrt {2}\right ) \end {gather*}

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]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\sqrt {2}+x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {2}+x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.11 \begin {gather*} \frac {1}{4} \left (-6-2 \sqrt {2} x^2+x^4+4 \log \left (\sqrt {2}+x^2\right )\right ) \end {gather*}

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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Mathics [A]
time = 1.79, size = 22, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {2} x^2}{2}+\frac {x^4}{4}+\text {Log}\left [\sqrt {2}+x^2\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 23, normalized size = 0.82

method	result	size
default	\(\frac {x^{4}}{4}+\ln \left (x^{2}+\sqrt {2}\right )-\frac {x^{2} \sqrt {2}}{2}\)	\(23\)
risch	\(\frac {x^{4}}{4}-\frac {x^{2} \sqrt {2}}{2}+\frac {1}{2}+\ln \left (x^{2}+\sqrt {2}\right )\)	\(24\)
meijerg	\(-\frac {x^{2} \sqrt {2}\, \left (-\frac {3 x^{2} \sqrt {2}}{2}+6\right )}{12}+\ln \left (1+\frac {x^{2} \sqrt {2}}{2}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^2+2^(1/2)),x,method=_RETURNVERBOSE)

[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 = 0.36, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {1}{2} \, \sqrt {2} x^{2} + \log \left (x^{2} + \sqrt {2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^2+2^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.31, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {1}{2} \, \sqrt {2} x^{2} + \log \left (x^{2} + \sqrt {2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^2+2^(1/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.07, size = 24, normalized size = 0.86 \begin {gather*} \frac {x^{4}}{4} - \frac {\sqrt {2} x^{2}}{2} + \log {\left (x^{2} + \sqrt {2} \right )} \end {gather*}

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 [A]
time = 0.00, size = 30, normalized size = 1.07 \begin {gather*} \frac {x^{4}-\left (2 \sqrt {2}\right ) x^{2}}{4}+\ln \left (x^{2}+\sqrt {2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^2+2^(1/2)),x)

[Out]

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

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Mupad [B]
time = 0.06, size = 22, normalized size = 0.79 \begin {gather*} \ln \left (x^2+\sqrt {2}\right )-\frac {\sqrt {2}\,x^2}{2}+\frac {x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(2^(1/2) + x^2),x)

[Out]

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

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