3.1385 \(\int \frac{x^{11}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{2 \sqrt{x^6+2}}{3} \]

[Out]

(-2*Sqrt[2 + x^6])/3 + (2 + x^6)^(3/2)/9

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Rubi [A]  time = 0.0109, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{2 \sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[2 + x^6],x]

[Out]

(-2*Sqrt[2 + x^6])/3 + (2 + x^6)^(3/2)/9

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^{11}}{\sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{\sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (-\frac{2}{\sqrt{2+x}}+\sqrt{2+x}\right ) \, dx,x,x^6\right )\\ &=-\frac{2}{3} \sqrt{2+x^6}+\frac{1}{9} \left (2+x^6\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0048161, size = 18, normalized size = 0.67 \[ \frac{1}{9} \left (x^6-4\right ) \sqrt{x^6+2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/Sqrt[2 + x^6],x]

[Out]

((-4 + x^6)*Sqrt[2 + x^6])/9

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Maple [A]  time = 0.003, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{6}-4}{9}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^6+2)^(1/2),x)

[Out]

1/9*(x^6+2)^(1/2)*(x^6-4)

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Maxima [A]  time = 1.002, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/9*(x^6 + 2)^(3/2) - 2/3*sqrt(x^6 + 2)

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Fricas [A]  time = 1.44149, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{9} \, \sqrt{x^{6} + 2}{\left (x^{6} - 4\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/9*sqrt(x^6 + 2)*(x^6 - 4)

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Sympy [A]  time = 1.13623, size = 24, normalized size = 0.89 \begin{align*} \frac{x^{6} \sqrt{x^{6} + 2}}{9} - \frac{4 \sqrt{x^{6} + 2}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**6+2)**(1/2),x)

[Out]

x**6*sqrt(x**6 + 2)/9 - 4*sqrt(x**6 + 2)/9

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Giac [A]  time = 1.10393, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

1/9*(x^6 + 2)^(3/2) - 2/3*sqrt(x^6 + 2)