3.14.95 \(\int \frac {1}{x^{16} \sqrt {2+x^6}} \, dx\) [1395]

Optimal. Leaf size=49 \[ -\frac {\sqrt {2+x^6}}{30 x^{15}}+\frac {\sqrt {2+x^6}}{45 x^9}-\frac {\sqrt {2+x^6}}{45 x^3} \]

[Out]

-1/30*(x^6+2)^(1/2)/x^15+1/45*(x^6+2)^(1/2)/x^9-1/45*(x^6+2)^(1/2)/x^3

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Rubi [A]
time = 0.01, antiderivative size = 49, 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 = {277, 270} \begin {gather*} -\frac {\sqrt {x^6+2}}{30 x^{15}}+\frac {\sqrt {x^6+2}}{45 x^9}-\frac {\sqrt {x^6+2}}{45 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

-1/30*Sqrt[2 + x^6]/x^15 + Sqrt[2 + x^6]/(45*x^9) - Sqrt[2 + x^6]/(45*x^3)

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*} \int \frac {1}{x^{16} \sqrt {2+x^6}} \, dx &=-\frac {\sqrt {2+x^6}}{30 x^{15}}-\frac {2}{5} \int \frac {1}{x^{10} \sqrt {2+x^6}} \, dx\\ &=-\frac {\sqrt {2+x^6}}{30 x^{15}}+\frac {\sqrt {2+x^6}}{45 x^9}+\frac {2}{15} \int \frac {1}{x^4 \sqrt {2+x^6}} \, dx\\ &=-\frac {\sqrt {2+x^6}}{30 x^{15}}+\frac {\sqrt {2+x^6}}{45 x^9}-\frac {\sqrt {2+x^6}}{45 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 28, normalized size = 0.57 \begin {gather*} \frac {\sqrt {2+x^6} \left (-3+2 x^6-2 x^{12}\right )}{90 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

(Sqrt[2 + x^6]*(-3 + 2*x^6 - 2*x^12))/(90*x^15)

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Maple [A]
time = 0.18, size = 25, normalized size = 0.51

method	result	size
gosper	\(-\frac {\sqrt {x^{6}+2}\, \left (2 x^{12}-2 x^{6}+3\right )}{90 x^{15}}\)	\(25\)
trager	\(-\frac {\sqrt {x^{6}+2}\, \left (2 x^{12}-2 x^{6}+3\right )}{90 x^{15}}\)	\(25\)
meijerg	\(-\frac {\sqrt {2}\, \left (\frac {2}{3} x^{12}-\frac {2}{3} x^{6}+1\right ) \sqrt {1+\frac {x^{6}}{2}}}{30 x^{15}}\)	\(30\)
risch	\(-\frac {2 x^{18}+2 x^{12}-x^{6}+6}{90 x^{15} \sqrt {x^{6}+2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(x^6+2)^(1/2)*(2*x^12-2*x^6+3)/x^15

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Maxima [A]
time = 0.29, size = 37, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {x^{6} + 2}}{24 \, x^{3}} + \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{36 \, x^{9}} - \frac {{\left (x^{6} + 2\right )}^{\frac {5}{2}}}{120 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(x^6 + 2)/x^3 + 1/36*(x^6 + 2)^(3/2)/x^9 - 1/120*(x^6 + 2)^(5/2)/x^15

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Fricas [A]
time = 0.36, size = 31, normalized size = 0.63 \begin {gather*} -\frac {2 \, x^{15} + {\left (2 \, x^{12} - 2 \, x^{6} + 3\right )} \sqrt {x^{6} + 2}}{90 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(2*x^15 + (2*x^12 - 2*x^6 + 3)*sqrt(x^6 + 2))/x^15

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Sympy [A]
time = 1.06, size = 41, normalized size = 0.84 \begin {gather*} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{45} + \frac {\sqrt {1 + \frac {2}{x^{6}}}}{45 x^{6}} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{30 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + 2/x**6)/45 + sqrt(1 + 2/x**6)/(45*x**6) - sqrt(1 + 2/x**6)/(30*x**12)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 1.24, size = 24, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {x^6+2}\,\left (2\,x^{12}-2\,x^6+3\right )}{90\,x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^16*(x^6 + 2)^(1/2)),x)

[Out]

-((x^6 + 2)^(1/2)*(2*x^12 - 2*x^6 + 3))/(90*x^15)

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