3.3.87 \(\int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx\)

Optimal. Leaf size=26 \[ \frac {\sqrt {x^6-1} \left (2 x^{12}+x^6-3\right )}{45 x^{15}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \begin {gather*} \frac {\left (x^6-1\right )^{3/2}}{15 x^{15}}+\frac {2 \left (x^6-1\right )^{3/2}}{45 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^6]/x^16,x]

[Out]

(-1 + x^6)^(3/2)/(15*x^15) + (2*(-1 + x^6)^(3/2))/(45*x^9)

Rule 264

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 271

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*} \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx &=\frac {\left (-1+x^6\right )^{3/2}}{15 x^{15}}+\frac {2}{5} \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx\\ &=\frac {\left (-1+x^6\right )^{3/2}}{15 x^{15}}+\frac {2 \left (-1+x^6\right )^{3/2}}{45 x^9}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\left (x^6-1\right )^{3/2} \left (2 x^6+3\right )}{45 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^6]/x^16,x]

[Out]

((-1 + x^6)^(3/2)*(3 + 2*x^6))/(45*x^15)

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IntegrateAlgebraic [A]  time = 0.13, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (-3+x^6+2 x^{12}\right )}{45 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x^6]/x^16,x]

[Out]

(Sqrt[-1 + x^6]*(-3 + x^6 + 2*x^12))/(45*x^15)

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fricas [A]  time = 0.47, size = 29, normalized size = 1.12 \begin {gather*} \frac {2 \, x^{15} + {\left (2 \, x^{12} + x^{6} - 3\right )} \sqrt {x^{6} - 1}}{45 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="fricas")

[Out]

1/45*(2*x^15 + (2*x^12 + x^6 - 3)*sqrt(x^6 - 1))/x^15

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giac [A]  time = 0.42, size = 41, normalized size = 1.58 \begin {gather*} -\frac {3 \, {\left (\frac {1}{x^{6}} - 1\right )}^{2} \sqrt {-\frac {1}{x^{6}} + 1} - 5 \, {\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}}}{45 \, \mathrm {sgn}\relax (x)} - \frac {2}{45} \, \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="giac")

[Out]

-1/45*(3*(1/x^6 - 1)^2*sqrt(-1/x^6 + 1) - 5*(-1/x^6 + 1)^(3/2))/sgn(x) - 2/45*sgn(x)

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maple [A]  time = 0.09, size = 23, normalized size = 0.88

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (2 x^{12}+x^{6}-3\right )}{45 x^{15}}\)	\(23\)
risch	\(\frac {2 x^{18}-x^{12}-4 x^{6}+3}{45 x^{15} \sqrt {x^{6}-1}}\)	\(30\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (2 x^{6}+3\right ) \sqrt {x^{6}-1}}{45 x^{15}}\)	\(40\)
meijerg	\(-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {2}{3} x^{12}-\frac {1}{3} x^{6}+1\right ) \sqrt {-x^{6}+1}}{15 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, x^{15}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/2)/x^16,x,method=_RETURNVERBOSE)

[Out]

1/45*(x^6-1)^(1/2)*(2*x^12+x^6-3)/x^15

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maxima [A]  time = 0.60, size = 25, normalized size = 0.96 \begin {gather*} \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {{\left (x^{6} - 1\right )}^{\frac {5}{2}}}{15 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="maxima")

[Out]

1/9*(x^6 - 1)^(3/2)/x^9 - 1/15*(x^6 - 1)^(5/2)/x^15

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mupad [B]  time = 0.39, size = 24, normalized size = 0.92 \begin {gather*} \frac {5\,{\left (x^6-1\right )}^{3/2}+2\,{\left (x^6-1\right )}^{5/2}}{45\,x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)^(1/2)/x^16,x)

[Out]

(5*(x^6 - 1)^(3/2) + 2*(x^6 - 1)^(5/2))/(45*x^15)

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sympy [A]  time = 1.37, size = 92, normalized size = 3.54 \begin {gather*} \begin {cases} \frac {2 \sqrt {x^{6} - 1}}{45 x^{3}} + \frac {\sqrt {x^{6} - 1}}{45 x^{9}} - \frac {\sqrt {x^{6} - 1}}{15 x^{15}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{6}}}{45 x^{3}} + \frac {i \sqrt {1 - x^{6}}}{45 x^{9}} - \frac {i \sqrt {1 - x^{6}}}{15 x^{15}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(1/2)/x**16,x)

[Out]

Piecewise((2*sqrt(x**6 - 1)/(45*x**3) + sqrt(x**6 - 1)/(45*x**9) - sqrt(x**6 - 1)/(15*x**15), Abs(x**6) > 1),
(2*I*sqrt(1 - x**6)/(45*x**3) + I*sqrt(1 - x**6)/(45*x**9) - I*sqrt(1 - x**6)/(15*x**15), True))

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