3.1419 \(\int \frac{1}{x^{10} (2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{x^3}{9 \sqrt{x^6+2}}+\frac{1}{9 \sqrt{x^6+2} x^3}-\frac{1}{18 \sqrt{x^6+2} x^9} \]

[Out]

-1/(18*x^9*Sqrt[2 + x^6]) + 1/(9*x^3*Sqrt[2 + x^6]) + x^3/(9*Sqrt[2 + x^6])

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Rubi [A]  time = 0.0103437, antiderivative size = 49, normalized size of antiderivative = 1., 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 = {271, 264} \[ \frac{x^3}{9 \sqrt{x^6+2}}+\frac{1}{9 \sqrt{x^6+2} x^3}-\frac{1}{18 \sqrt{x^6+2} x^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

-1/(18*x^9*Sqrt[2 + x^6]) + 1/(9*x^3*Sqrt[2 + x^6]) + x^3/(9*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^{10} \left (2+x^6\right )^{3/2}} \, dx &=-\frac{1}{18 x^9 \sqrt{2+x^6}}-\frac{2}{3} \int \frac{1}{x^4 \left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac{1}{18 x^9 \sqrt{2+x^6}}+\frac{1}{9 x^3 \sqrt{2+x^6}}+\frac{2}{3} \int \frac{x^2}{\left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac{1}{18 x^9 \sqrt{2+x^6}}+\frac{1}{9 x^3 \sqrt{2+x^6}}+\frac{x^3}{9 \sqrt{2+x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0056997, size = 28, normalized size = 0.57 \[ \frac{2 x^{12}+2 x^6-1}{18 x^9 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

(-1 + 2*x^6 + 2*x^12)/(18*x^9*Sqrt[2 + x^6])

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*}{\frac{2\,{x}^{12}+2\,{x}^{6}-1}{18\,{x}^{9}}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(x^6+2)^(3/2),x)

[Out]

1/18*(2*x^12+2*x^6-1)/x^9/(x^6+2)^(1/2)

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Maxima [A]  time = 1.0096, size = 50, normalized size = 1.02 \begin{align*} \frac{x^{3}}{24 \, \sqrt{x^{6} + 2}} + \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{72 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

1/24*x^3/sqrt(x^6 + 2) + 1/12*sqrt(x^6 + 2)/x^3 - 1/72*(x^6 + 2)^(3/2)/x^9

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Fricas [A]  time = 1.48529, size = 101, normalized size = 2.06 \begin{align*} \frac{2 \, x^{15} + 4 \, x^{9} +{\left (2 \, x^{12} + 2 \, x^{6} - 1\right )} \sqrt{x^{6} + 2}}{18 \,{\left (x^{15} + 2 \, x^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/18*(2*x^15 + 4*x^9 + (2*x^12 + 2*x^6 - 1)*sqrt(x^6 + 2))/(x^15 + 2*x^9)

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Sympy [A]  time = 1.87084, size = 70, normalized size = 1.43 \begin{align*} \frac{2 x^{12} \sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} + \frac{2 x^{6} \sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(x**6+2)**(3/2),x)

[Out]

2*x**12*sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6) + 2*x**6*sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6) - sqrt(1 + 2/x**6
)/(18*x**12 + 36*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^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^10), x)