3.15.11 \(\int \frac {x^5}{(2+x^6)^{3/2}} \, dx\) [1411]

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

[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 = {267} \begin {gather*} -\frac {1}{3 \sqrt {x^6+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(2 + x^6)^(3/2),x]

[Out]

-1/3*1/Sqrt[2 + x^6]

Rule 267

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 = 0.02, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{3 \sqrt {2+x^6}} \end {gather*}

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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Maple [A]
time = 0.17, size = 10, normalized size = 0.77

method	result	size
gosper	\(-\frac {1}{3 \sqrt {x^{6}+2}}\)	\(10\)
derivativedivides	\(-\frac {1}{3 \sqrt {x^{6}+2}}\)	\(10\)
default	\(-\frac {1}{3 \sqrt {x^{6}+2}}\)	\(10\)
trager	\(-\frac {1}{3 \sqrt {x^{6}+2}}\)	\(10\)
risch	\(-\frac {1}{3 \sqrt {x^{6}+2}}\)	\(10\)
meijerg	\(\frac {\sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {x^{6}}{2}}}\right )}{6 \sqrt {\pi }}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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 = 0.36, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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 = 0.17, size = 12, normalized size = 0.92 \begin {gather*} - \frac {1}{3 \sqrt {x^{6} + 2}} \end {gather*}

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 = 1.17, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3 \, \sqrt {x^{6} + 2}} \end {gather*}

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

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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