3.120 \(\int \frac {x}{\sqrt {4+x^2}} \, dx\)

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

[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 = {261} \[ \sqrt {x^2+4} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^2]

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}{\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 \[ \sqrt {x^2+4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^2]

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fricas [A]  time = 0.41, size = 7, normalized size = 0.78 \[ \sqrt {x^{2} + 4} \]

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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giac [A]  time = 0.89, size = 7, normalized size = 0.78 \[ \sqrt {x^{2} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 4)

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maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ \sqrt {x^{2}+4} \]

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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maxima [A]  time = 0.43, size = 7, normalized size = 0.78 \[ \sqrt {x^{2} + 4} \]

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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mupad [B]  time = 0.03, size = 7, normalized size = 0.78 \[ \sqrt {x^2+4} \]

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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sympy [A]  time = 0.15, size = 7, normalized size = 0.78 \[ \sqrt {x^{2} + 4} \]

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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