3.6.58 \(\int \frac {x}{\sqrt {-9+4 x^2}} \, dx\) [558]

Optimal. Leaf size=15 \[ \frac {1}{4} \sqrt {-9+4 x^2} \]

[Out]

1/4*(4*x^2-9)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 15, 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}{4} \sqrt {4 x^2-9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-9 + 4*x^2],x]

[Out]

Sqrt[-9 + 4*x^2]/4

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 {-9+4 x^2}} \, dx &=\frac {1}{4} \sqrt {-9+4 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt {-9+4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-9 + 4*x^2],x]

[Out]

Sqrt[-9 + 4*x^2]/4

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Maple [A]
time = 0.05, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {\sqrt {4 x^{2}-9}}{4}\)	\(12\)
default	\(\frac {\sqrt {4 x^{2}-9}}{4}\)	\(12\)
trager	\(\frac {\sqrt {4 x^{2}-9}}{4}\)	\(12\)
risch	\(\frac {\sqrt {4 x^{2}-9}}{4}\)	\(12\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right )}{4 \sqrt {4 x^{2}-9}}\)	\(22\)
meijerg	\(-\frac {3 \sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-\frac {4 x^{2}}{9}}\right )}{8 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x^2-9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(4*x^2-9)^(1/2)

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Maxima [A]
time = 0.28, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, \sqrt {4 \, x^{2} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(4*x^2 - 9)

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Fricas [A]
time = 1.33, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, \sqrt {4 \, x^{2} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(4*x^2 - 9)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.67 \begin {gather*} \frac {\sqrt {4 x^{2} - 9}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x**2-9)**(1/2),x)

[Out]

sqrt(4*x**2 - 9)/4

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Giac [A]
time = 0.90, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, \sqrt {4 \, x^{2} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(4*x^2 - 9)

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Mupad [B]
time = 0.14, size = 11, normalized size = 0.73 \begin {gather*} \frac {\sqrt {4\,x^2-9}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x^2 - 9)^(1/2),x)

[Out]

(4*x^2 - 9)^(1/2)/4

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