\(\int \frac {1}{x^{10} (2+x^6)^{3/2}} \, dx\) [1419]

Optimal result

Integrand size = 13, antiderivative size = 49 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=-\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {x^3}{9 \sqrt {2+x^6}} \]

[Out]

-1/18/x^9/(x^6+2)^(1/2)+1/9/x^3/(x^6+2)^(1/2)+1/9*x^3/(x^6+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, 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 = {277, 270} \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=-\frac {1}{18 \sqrt {x^6+2} x^9}+\frac {x^3}{9 \sqrt {x^6+2}}+\frac {1}{9 \sqrt {x^6+2} x^3} \]

[In]

Int[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

-1/18*1/(x^9*Sqrt[2 + x^6]) + 1/(9*x^3*Sqrt[2 + x^6]) + x^3/(9*Sqrt[2 + x^6])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{18 x^9 \sqrt {2+x^6}}-\frac {2}{3} \int \frac {1}{x^4 \left (2+x^6\right )^{3/2}} \, dx \\ & = -\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {2}{3} \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx \\ & = -\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {x^3}{9 \sqrt {2+x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=\frac {-1+2 x^6+2 x^{12}}{18 x^9 \sqrt {2+x^6}} \]

[In]

Integrate[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

(-1 + 2*x^6 + 2*x^12)/(18*x^9*Sqrt[2 + x^6])

Maple [A] (verified)

Time = 4.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51

method	result	size
gosper	\(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\)	\(25\)
trager	\(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\)	\(25\)
risch	\(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\)	\(25\)
pseudoelliptic	\(\frac {2 x^{12}+2 x^{6}-1}{18 x^{9} \sqrt {x^{6}+2}}\)	\(25\)
meijerg	\(-\frac {\sqrt {2}\, \left (-2 x^{12}-2 x^{6}+1\right )}{36 x^{9} \sqrt {1+\frac {x^{6}}{2}}}\)	\(30\)

[In]

int(1/x^10/(x^6+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/18*(2*x^12+2*x^6-1)/x^9/(x^6+2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=\frac {2 \, x^{15} + 4 \, x^{9} + {\left (2 \, x^{12} + 2 \, x^{6} - 1\right )} \sqrt {x^{6} + 2}}{18 \, {\left (x^{15} + 2 \, x^{9}\right )}} \]

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/18*(2*x^15 + 4*x^9 + (2*x^12 + 2*x^6 - 1)*sqrt(x^6 + 2))/(x^15 + 2*x^9)

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=\frac {2 x^{12} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} + \frac {2 x^{6} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} \]

[In]

integrate(1/x**10/(x**6+2)**(3/2),x)

[Out]

2*x**12*sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6) + 2*x**6*sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6) - sqrt(1 + 2/x**6
)/(18*x**12 + 36*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=\frac {x^{3}}{24 \, \sqrt {x^{6} + 2}} + \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{72 \, x^{9}} \]

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

1/24*x^3/sqrt(x^6 + 2) + 1/12*sqrt(x^6 + 2)/x^3 - 1/72*(x^6 + 2)^(3/2)/x^9

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 5.63 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx=-\frac {24\,x^6-8\,{\left (x^6+2\right )}^2+36}{72\,x^9\,\sqrt {x^6+2}} \]

[In]

int(1/(x^10*(x^6 + 2)^(3/2)),x)

[Out]

-(24*x^6 - 8*(x^6 + 2)^2 + 36)/(72*x^9*(x^6 + 2)^(1/2))