3.2.39 \(\int \frac {x}{(4+x^2)^{5/2}} \, dx\) [139]

Optimal. Leaf size=13 \[ -\frac {1}{3 \left (4+x^2\right )^{3/2}} \]

[Out]

-1/3/(x^2+4)^(3/2)

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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \begin {gather*} -\frac {1}{3 \left (x^2+4\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(4 + x^2)^(5/2),x]

[Out]

-1/3*1/(4 + x^2)^(3/2)

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}{\left (4+x^2\right )^{5/2}} \, dx &=-\frac {1}{3 \left (4+x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{3 \left (4+x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(4 + x^2)^(5/2),x]

[Out]

-1/3*1/(4 + x^2)^(3/2)

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Maple [A]
time = 0.04, size = 10, normalized size = 0.77

method	result	size
gosper	\(-\frac {1}{3 \left (x^{2}+4\right )^{\frac {3}{2}}}\)	\(10\)
derivativedivides	\(-\frac {1}{3 \left (x^{2}+4\right )^{\frac {3}{2}}}\)	\(10\)
default	\(-\frac {1}{3 \left (x^{2}+4\right )^{\frac {3}{2}}}\)	\(10\)
trager	\(-\frac {1}{3 \left (x^{2}+4\right )^{\frac {3}{2}}}\)	\(10\)
risch	\(-\frac {1}{3 \left (x^{2}+4\right )^{\frac {3}{2}}}\)	\(10\)
meijerg	\(\frac {\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1+\frac {x^{2}}{4}\right )^{\frac {3}{2}}}}{12 \sqrt {\pi }}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+4)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(x^2+4)^(3/2)

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Maxima [A]
time = 1.22, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3 \, {\left (x^{2} + 4\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(5/2),x, algorithm="maxima")

[Out]

-1/3/(x^2 + 4)^(3/2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
time = 0.54, size = 21, normalized size = 1.62 \begin {gather*} -\frac {\sqrt {x^{2} + 4}}{3 \, {\left (x^{4} + 8 \, x^{2} + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(5/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(x^2 + 4)/(x^4 + 8*x^2 + 16)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.48, size = 26, normalized size = 2.00 \begin {gather*} - \frac {1}{3 x^{2} \sqrt {x^{2} + 4} + 12 \sqrt {x^{2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+4)**(5/2),x)

[Out]

-1/(3*x**2*sqrt(x**2 + 4) + 12*sqrt(x**2 + 4))

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Giac [A]
time = 0.57, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3 \, {\left (x^{2} + 4\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(5/2),x, algorithm="giac")

[Out]

-1/3/(x^2 + 4)^(3/2)

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Mupad [B]
time = 0.16, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{3\,{\left (x^2+4\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 + 4)^(5/2),x)

[Out]

-1/(3*(x^2 + 4)^(3/2))

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