3.1411 \(\int \frac{x^5}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{3 \sqrt{x^6+2}} \]

[Out]

-1/(3*Sqrt[2 + x^6])

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Rubi [A]  time = 0.002953, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(2 + x^6)^(3/2),x]

[Out]

-1/(3*Sqrt[2 + x^6])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0020231, size = 13, normalized size = 1. \[ -\frac{1}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(2 + x^6)^(3/2),x]

[Out]

-1/(3*Sqrt[2 + x^6])

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Maple [A]  time = 0.004, size = 10, normalized size = 0.8 \begin{align*} -{\frac{1}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.998981, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

-1/3/sqrt(x^6 + 2)

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Fricas [A]  time = 1.45042, size = 27, normalized size = 2.08 \begin{align*} -\frac{1}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

-1/3/sqrt(x^6 + 2)

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Sympy [A]  time = 0.445161, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{3 \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**6+2)**(3/2),x)

[Out]

-1/(3*sqrt(x**6 + 2))

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Giac [A]  time = 1.1642, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

-1/3/sqrt(x^6 + 2)