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

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

[Out]

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

________________________________________________________________________________________

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

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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-9 - 4*x^2],x]

[Out]

-1/4*Sqrt[-9 - 4*x^2]

________________________________________________________________________________________

fricas [A]  time = 0.69, 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 12, normalized size = 0.80 \[ -\frac {\sqrt {-4 x^{2}-9}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, 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)

________________________________________________________________________________________

mupad [B]  time = 4.99, size = 11, normalized size = 0.73 \[ -\frac {\sqrt {-4\,x^2-9}}{4} \]

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

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 14, normalized size = 0.93 \[ - \frac {\sqrt {- 4 x^{2} - 9}}{4} \]

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

________________________________________________________________________________________