3.338 \(\int \frac{x^2}{\sqrt{5-4 x^2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{5}{16} \sin ^{-1}\left (\frac{2 x}{\sqrt{5}}\right )-\frac{1}{8} x \sqrt{5-4 x^2} \]

[Out]

-(x*Sqrt[5 - 4*x^2])/8 + (5*ArcSin[(2*x)/Sqrt[5]])/16

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Rubi [A]  time = 0.0211762, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{5}{16} \sin ^{-1}\left (\frac{2 x}{\sqrt{5}}\right )-\frac{1}{8} x \sqrt{5-4 x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[5 - 4*x^2],x]

[Out]

-(x*Sqrt[5 - 4*x^2])/8 + (5*ArcSin[(2*x)/Sqrt[5]])/16

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Rubi in Sympy [A]  time = 1.78426, size = 27, normalized size = 0.9 \[ - \frac{x \sqrt{- 4 x^{2} + 5}}{8} + \frac{5 \operatorname{asin}{\left (\frac{2 \sqrt{5} x}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-4*x**2+5)**(1/2),x)

[Out]

-x*sqrt(-4*x**2 + 5)/8 + 5*asin(2*sqrt(5)*x/5)/16

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Mathematica [A]  time = 0.0185408, size = 30, normalized size = 1. \[ \frac{5}{16} \sin ^{-1}\left (\frac{2 x}{\sqrt{5}}\right )-\frac{1}{8} x \sqrt{5-4 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[5 - 4*x^2],x]

[Out]

-(x*Sqrt[5 - 4*x^2])/8 + (5*ArcSin[(2*x)/Sqrt[5]])/16

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Maple [A]  time = 0.008, size = 23, normalized size = 0.8 \[{\frac{5}{16}\arcsin \left ({\frac{2\,x\sqrt{5}}{5}} \right ) }-{\frac{x}{8}\sqrt{-4\,{x}^{2}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-4*x^2+5)^(1/2),x)

[Out]

5/16*arcsin(2/5*x*5^(1/2))-1/8*x*(-4*x^2+5)^(1/2)

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Maxima [A]  time = 1.52561, size = 30, normalized size = 1. \[ -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x + \frac{5}{16} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-4*x^2 + 5),x, algorithm="maxima")

[Out]

-1/8*sqrt(-4*x^2 + 5)*x + 5/16*arcsin(2/5*sqrt(5)*x)

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Fricas [A]  time = 0.212454, size = 41, normalized size = 1.37 \[ -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x - \frac{5}{16} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 5}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-4*x^2 + 5),x, algorithm="fricas")

[Out]

-1/8*sqrt(-4*x^2 + 5)*x - 5/16*arctan(1/2*sqrt(-4*x^2 + 5)/x)

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Sympy [A]  time = 0.268908, size = 27, normalized size = 0.9 \[ - \frac{x \sqrt{- 4 x^{2} + 5}}{8} + \frac{5 \operatorname{asin}{\left (\frac{2 \sqrt{5} x}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-4*x**2+5)**(1/2),x)

[Out]

-x*sqrt(-4*x**2 + 5)/8 + 5*asin(2*sqrt(5)*x/5)/16

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GIAC/XCAS [A]  time = 0.220797, size = 30, normalized size = 1. \[ -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x + \frac{5}{16} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-4*x^2 + 5),x, algorithm="giac")

[Out]

-1/8*sqrt(-4*x^2 + 5)*x + 5/16*arcsin(2/5*sqrt(5)*x)