\(\int \frac {x^{11}}{\sqrt {2+x^6}} \, dx\) [1385]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=-\frac {2}{3} \sqrt {2+x^6}+\frac {1}{9} \left (2+x^6\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {1}{9} \left (x^6+2\right )^{3/2}-\frac {2 \sqrt {x^6+2}}{3} \]

[In]

Int[x^11/Sqrt[2 + x^6],x]

[Out]

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

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*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {x}{\sqrt {2+x}} \, dx,x,x^6\right ) \\ & = \frac {1}{6} \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {1}{9} \left (-4+x^6\right ) \sqrt {2+x^6} \]

[In]

Integrate[x^11/Sqrt[2 + x^6],x]

[Out]

((-4 + x^6)*Sqrt[2 + x^6])/9

Maple [A] (verified)

Time = 4.58 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
gosper	\(\frac {\sqrt {x^{6}+2}\, \left (x^{6}-4\right )}{9}\)	\(15\)
risch	\(\frac {\sqrt {x^{6}+2}\, \left (x^{6}-4\right )}{9}\)	\(15\)
pseudoelliptic	\(\frac {\sqrt {x^{6}+2}\, \left (x^{6}-4\right )}{9}\)	\(15\)
trager	\(\sqrt {x^{6}+2}\, \left (\frac {x^{6}}{9}-\frac {4}{9}\right )\)	\(16\)
meijerg	\(\frac {\sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 x^{6}+8\right ) \sqrt {1+\frac {x^{6}}{2}}}{6}\right )}{3 \sqrt {\pi }}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {1}{9} \, \sqrt {x^{6} + 2} {\left (x^{6} - 4\right )} \]

[In]

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

[Out]

1/9*sqrt(x^6 + 2)*(x^6 - 4)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {x^{6} \sqrt {x^{6} + 2}}{9} - \frac {4 \sqrt {x^{6} + 2}}{9} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {1}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {x^{6} + 2} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {1}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {x^{6} + 2} \]

[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)

Mupad [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^{11}}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {x^6+2}\,\left (x^6-4\right )}{9} \]

[In]

int(x^11/(x^6 + 2)^(1/2),x)

[Out]

((x^6 + 2)^(1/2)*(x^6 - 4))/9