\(\int \frac {\sqrt {1+x^6}}{x^{10}} \, dx\) [106]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {\left (1+x^6\right )^{3/2}}{9 x^9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {\left (x^6+1\right )^{3/2}}{9 x^9} \]

[In]

Int[Sqrt[1 + x^6]/x^10,x]

[Out]

-1/9*(1 + x^6)^(3/2)/x^9

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]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {\left (1+x^6\right )^{3/2}}{9 x^9} \]

[In]

Integrate[Sqrt[1 + x^6]/x^10,x]

[Out]

-1/9*(1 + x^6)^(3/2)/x^9

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(-\frac {\left (x^{6}+1\right )^{\frac {3}{2}}}{9 x^{9}}\)	\(13\)
meijerg	\(-\frac {\left (x^{6}+1\right )^{\frac {3}{2}}}{9 x^{9}}\)	\(13\)
pseudoelliptic	\(-\frac {\left (x^{6}+1\right )^{\frac {3}{2}}}{9 x^{9}}\)	\(13\)
risch	\(-\frac {x^{12}+2 x^{6}+1}{9 x^{9} \sqrt {x^{6}+1}}\)	\(23\)
gosper	\(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \sqrt {x^{6}+1}}{9 x^{9}}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {x^{9} + {\left (x^{6} + 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]

[In]

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

[Out]

-1/9*(x^9 + (x^6 + 1)^(3/2))/x^9

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=- \frac {\sqrt {1 + \frac {1}{x^{6}}}}{9} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{9 x^{6}} \]

[In]

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

[Out]

-sqrt(1 + x**(-6))/9 - sqrt(1 + x**(-6))/(9*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {{\left (x^{6} + 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]

[In]

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

[Out]

-1/9*(x^6 + 1)^(3/2)/x^9

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {{\left (\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}}}{9 \, \mathrm {sgn}\left (x\right )} + \frac {1}{9} \, \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1+x^6}}{x^{10}} \, dx=-\frac {{\left (x^6+1\right )}^{3/2}}{9\,x^9} \]

[In]

int((x^6 + 1)^(1/2)/x^10,x)

[Out]

-(x^6 + 1)^(3/2)/(9*x^9)