### 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 = 0.0191322, 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.

[In]

Int[x^5/(Sqrt[2] + x^2),x]

[Out]

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

Rule 266

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]]

Rule 43

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])

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.

[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 = 0.007, 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.

[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 = 1.44337, 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.

[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 = 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.

[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 = 0.114682, 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.

[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 = 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.

[In]

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

[Out]

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