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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{3 \sqrt {x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(2 + x^6)^(3/2),x]

[Out]

-1/3*1/Sqrt[2 + x^6]

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fricas [A]  time = 0.89, size = 9, normalized size = 0.69 \[ -\frac {1}{3 \, \sqrt {x^{6} + 2}} \]

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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giac [A]  time = 0.17, size = 9, normalized size = 0.69 \[ -\frac {1}{3 \, \sqrt {x^{6} + 2}} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ -\frac {1}{3 \sqrt {x^{6}+2}} \]

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 = 1.04, size = 9, normalized size = 0.69 \[ -\frac {1}{3 \, \sqrt {x^{6} + 2}} \]

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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mupad [B]  time = 1.15, size = 9, normalized size = 0.69 \[ -\frac {1}{3\,\sqrt {x^6+2}} \]

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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sympy [A]  time = 0.61, size = 12, normalized size = 0.92 \[ - \frac {1}{3 \sqrt {x^{6} + 2}} \]

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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