3.15.18 \(\int \frac {1}{x^4 (2+x^6)^{3/2}} \, dx\) [1418]

Optimal. Leaf size=33 \[ -\frac {1}{6 x^3 \sqrt {2+x^6}}-\frac {x^3}{6 \sqrt {2+x^6}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} -\frac {x^3}{6 \sqrt {x^6+2}}-\frac {1}{6 \sqrt {x^6+2} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(2 + x^6)^(3/2)),x]

[Out]

-1/6*1/(x^3*Sqrt[2 + x^6]) - x^3/(6*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*} \int \frac {1}{x^4 \left (2+x^6\right )^{3/2}} \, dx &=-\frac {1}{6 x^3 \sqrt {2+x^6}}-\int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{6 x^3 \sqrt {2+x^6}}-\frac {x^3}{6 \sqrt {2+x^6}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 23, normalized size = 0.70 \begin {gather*} \frac {-1-x^6}{6 x^3 \sqrt {2+x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(2 + x^6)^(3/2)),x]

[Out]

(-1 - x^6)/(6*x^3*Sqrt[2 + x^6])

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Maple [A]
time = 0.17, size = 18, normalized size = 0.55

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 25, normalized size = 0.76 \begin {gather*} -\frac {x^{3}}{12 \, \sqrt {x^{6} + 2}} - \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x^{9} + 2 \, x^{3} + \sqrt {x^{6} + 2} {\left (x^{6} + 1\right )}}{6 \, {\left (x^{9} + 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.41, size = 31, normalized size = 0.94 \begin {gather*} - \frac {1}{6 \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{6 x^{6} \sqrt {1 + \frac {2}{x^{6}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(x**6+2)**(3/2),x)

[Out]

-1/(6*sqrt(1 + 2/x**6)) - 1/(6*x**6*sqrt(1 + 2/x**6))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^4), x)

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Mupad [B]
time = 1.15, size = 17, normalized size = 0.52 \begin {gather*} -\frac {x^6+1}{6\,x^3\,\sqrt {x^6+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^6 + 2)^(3/2)),x)

[Out]

-(x^6 + 1)/(6*x^3*(x^6 + 2)^(1/2))

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