3.1418 \(\int \frac{1}{x^4 (2+x^6)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0062853, antiderivative size = 33, normalized size of antiderivative = 1., 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} \[ -\frac{x^3}{6 \sqrt{x^6+2}}-\frac{1}{6 \sqrt{x^6+2} x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(2 + x^6)^(3/2)),x]

[Out]

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

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]

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]

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*}

Mathematica [A]  time = 0.004974, size = 23, normalized size = 0.7 \[ \frac{-x^6-1}{6 x^3 \sqrt{x^6+2}} \]

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 = 0.005, size = 18, normalized size = 0.6 \begin{align*} -{\frac{{x}^{6}+1}{6\,{x}^{3}}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985953, size = 34, normalized size = 1.03 \begin{align*} -\frac{x^{3}}{12 \, \sqrt{x^{6} + 2}} - \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} \end{align*}

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 = 1.44519, size = 81, normalized size = 2.45 \begin{align*} -\frac{x^{9} + 2 \, x^{3} + \sqrt{x^{6} + 2}{\left (x^{6} + 1\right )}}{6 \,{\left (x^{9} + 2 \, x^{3}\right )}} \end{align*}

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 = 0.798994, size = 31, normalized size = 0.94 \begin{align*} - \frac{1}{6 \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{6 x^{6} \sqrt{1 + \frac{2}{x^{6}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

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)