3.547 \(\int \frac {x}{\sqrt {9-4 x^2}} \, dx\)

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 = {261} \[ -\frac {1}{4} \sqrt {9-4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[9 - 4*x^2],x]

[Out]

-Sqrt[9 - 4*x^2]/4

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 {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 \[ -\frac {1}{4} \sqrt {9-4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[9 - 4*x^2],x]

[Out]

-1/4*Sqrt[9 - 4*x^2]

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fricas [A]  time = 0.98, size = 11, normalized size = 0.73 \[ -\frac {1}{4} \, \sqrt {-4 \, x^{2} + 9} \]

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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giac [A]  time = 1.13, size = 11, normalized size = 0.73 \[ -\frac {1}{4} \, \sqrt {-4 \, x^{2} + 9} \]

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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maple [A]  time = 0.00, size = 22, normalized size = 1.47 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right )}{4 \sqrt {-4 x^{2}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-4*x^2+9)^(1/2),x)

[Out]

1/4*(2*x-3)*(2*x+3)/(-4*x^2+9)^(1/2)

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maxima [A]  time = 1.33, size = 11, normalized size = 0.73 \[ -\frac {1}{4} \, \sqrt {-4 \, x^{2} + 9} \]

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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mupad [B]  time = 4.56, size = 11, normalized size = 0.73 \[ -\frac {\sqrt {\frac {9}{4}-x^2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(9 - 4*x^2)^(1/2),x)

[Out]

-(9/4 - x^2)^(1/2)/2

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sympy [A]  time = 0.15, size = 12, normalized size = 0.80 \[ - \frac {\sqrt {9 - 4 x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(9 - 4*x**2)/4

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