3.2.20 \(\int \frac {x}{\sqrt {4+x^2}} \, dx\) [120]

Optimal. Leaf size=9 \[ \sqrt {4+x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 9, 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*} \sqrt {x^2+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^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}{\sqrt {4+x^2}} \, dx &=\sqrt {4+x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \sqrt {4+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^2]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 1.65, size = 8, normalized size = 0.89 \begin {gather*} \text {ConditionalExpression}\left [4+x^2,\left \{\text {True}\right \}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x/Sqrt[4 + x^2],x]')

[Out]

ConditionalExpression[4 + x ^ 2, {True}]

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Maple [A]
time = 0.04, size = 8, normalized size = 0.89

method	result	size
gosper	\(\sqrt {x^{2}+4}\)	\(8\)
derivativedivides	\(\sqrt {x^{2}+4}\)	\(8\)
default	\(\sqrt {x^{2}+4}\)	\(8\)
trager	\(\sqrt {x^{2}+4}\)	\(8\)
risch	\(\sqrt {x^{2}+4}\)	\(8\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {x^{2}}{4}}}{\sqrt {\pi }}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 + 4)

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Fricas [A]
time = 0.33, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^2 + 4)

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Sympy [A]
time = 0.06, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+4)**(1/2),x)

[Out]

sqrt(x**2 + 4)

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Giac [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \sqrt {x^{2}+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(1/2),x)

[Out]

sqrt(x^2 + 4)

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Mupad [B]
time = 0.03, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {x^2+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 + 4)^(1/2),x)

[Out]

(x^2 + 4)^(1/2)

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