3.1410 \(\int \frac{x^{11}}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{x^6+2}}{3}+\frac{2}{3 \sqrt{x^6+2}} \]

[Out]

2/(3*Sqrt[2 + x^6]) + Sqrt[2 + x^6]/3

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Rubi [A]  time = 0.0108826, 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} \[ \frac{\sqrt{x^6+2}}{3}+\frac{2}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(2 + x^6)^(3/2),x]

[Out]

2/(3*Sqrt[2 + x^6]) + Sqrt[2 + x^6]/3

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

Mathematica [A]  time = 0.0053689, size = 18, normalized size = 0.67 \[ \frac{x^6+4}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(2 + x^6)^(3/2),x]

[Out]

(4 + x^6)/(3*Sqrt[2 + x^6])

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Maple [A]  time = 0.003, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{6}+4}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^6+2)^(3/2),x)

[Out]

1/3*(x^6+4)/(x^6+2)^(1/2)

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Maxima [A]  time = 0.983694, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{3} \, \sqrt{x^{6} + 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)^(3/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 2) + 2/3/sqrt(x^6 + 2)

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Fricas [A]  time = 1.41825, size = 39, normalized size = 1.44 \begin{align*} \frac{x^{6} + 4}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(x^6 + 4)/sqrt(x^6 + 2)

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Sympy [A]  time = 1.37388, size = 24, normalized size = 0.89 \begin{align*} \frac{x^{6}}{3 \sqrt{x^{6} + 2}} + \frac{4}{3 \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**6+2)**(3/2),x)

[Out]

x**6/(3*sqrt(x**6 + 2)) + 4/(3*sqrt(x**6 + 2))

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Giac [A]  time = 1.14739, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{3} \, \sqrt{x^{6} + 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)^(3/2),x, algorithm="giac")

[Out]

1/3*sqrt(x^6 + 2) + 2/3/sqrt(x^6 + 2)