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

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

[Out]

Sqrt[-9 + 4*x^2]/4

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Rubi [A]  time = 0.0087077, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{4} \sqrt{4 x^2-9} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-9 + 4*x^2],x]

[Out]

Sqrt[-9 + 4*x^2]/4

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Rubi in Sympy [A]  time = 1.96512, size = 10, normalized size = 0.67 \[ \frac{\sqrt{4 x^{2} - 9}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(4*x**2-9)**(1/2),x)

[Out]

sqrt(4*x**2 - 9)/4

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Mathematica [A]  time = 0.00256402, size = 15, normalized size = 1. \[ \frac{1}{4} \sqrt{4 x^2-9} \]

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 = 0.004, size = 22, normalized size = 1.5 \[{\frac{ \left ( 2\,x-3 \right ) \left ( 2\,x+3 \right ) }{4}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \]

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 = 1.34712, size = 15, normalized size = 1. \[ \frac{1}{4} \, \sqrt{4 \, x^{2} - 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(4*x^2 - 9),x, algorithm="maxima")

[Out]

1/4*sqrt(4*x^2 - 9)

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Fricas [A]  time = 0.214052, size = 51, normalized size = 3.4 \[ -\frac{4 \, x^{2} - 2 \, \sqrt{4 \, x^{2} - 9} x - 9}{4 \,{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(4*x^2 - 9),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.310601, size = 10, normalized size = 0.67 \[ \frac{\sqrt{4 x^{2} - 9}}{4} \]

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

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GIAC/XCAS [A]  time = 0.215375, size = 15, normalized size = 1. \[ \frac{1}{4} \, \sqrt{4 \, x^{2} - 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(4*x^2 - 9),x, algorithm="giac")

[Out]

1/4*sqrt(4*x^2 - 9)