3.15.10 \(\int \frac {x^{11}}{(2+x^6)^{3/2}} \, dx\) [1410]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \frac {\sqrt {x^6+2}}{3}+\frac {2}{3 \sqrt {x^6+2}} \end {gather*}

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 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^{11}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {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 = 0.02, size = 18, normalized size = 0.67 \begin {gather*} \frac {4+x^6}{3 \sqrt {2+x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11/(2 + x^6)^(3/2),x]

[Out]

(4 + x^6)/(3*Sqrt[2 + x^6])

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 15, normalized size = 0.56

method	result	size
gosper	\(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\)	\(15\)
trager	\(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\)	\(15\)
risch	\(\frac {x^{6}+4}{3 \sqrt {x^{6}+2}}\)	\(15\)
meijerg	\(\frac {\sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (2 x^{6}+8\right )}{4 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 14, normalized size = 0.52 \begin {gather*} \frac {x^{6} + 4}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.33, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{6}}{3 \sqrt {x^{6} + 2}} + \frac {4}{3 \sqrt {x^{6} + 2}} \end {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 1.25, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 2} + \frac {2}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 1.15, size = 14, normalized size = 0.52 \begin {gather*} \frac {x^6+4}{3\,\sqrt {x^6+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^6 + 2)^(3/2),x)

[Out]

(x^6 + 4)/(3*(x^6 + 2)^(1/2))

________________________________________________________________________________________