3.959 \(\int \frac{x^5}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=29 \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

[Out]

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

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Rubi [A]  time = 0.0420893, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[16 - x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.26836, size = 20, normalized size = 0.69 \[ - \frac{x^{2} \sqrt{- x^{4} + 16}}{4} + 4 \operatorname{asin}{\left (\frac{x^{2}}{4} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**4+16)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0163204, size = 29, normalized size = 1. \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[16 - x^4],x]

[Out]

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

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Maple [A]  time = 0.018, size = 24, normalized size = 0.8 \[ 4\,\arcsin \left ( 1/4\,{x}^{2} \right ) -{\frac{{x}^{2}}{4}\sqrt{-{x}^{4}+16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^4+16)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.58885, size = 59, normalized size = 2.03 \[ \frac{4 \, \sqrt{-x^{4} + 16}}{x^{2}{\left (\frac{x^{4} - 16}{x^{4}} - 1\right )}} - 4 \, \arctan \left (\frac{\sqrt{-x^{4} + 16}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 16),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.267674, size = 115, normalized size = 3.97 \[ \frac{8 \, x^{6} - 128 \, x^{2} - 32 \,{\left (x^{4} + 8 \, \sqrt{-x^{4} + 16} - 32\right )} \arctan \left (\frac{\sqrt{-x^{4} + 16} - 4}{x^{2}}\right ) -{\left (x^{6} - 32 \, x^{2}\right )} \sqrt{-x^{4} + 16}}{4 \,{\left (x^{4} + 8 \, \sqrt{-x^{4} + 16} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 16),x, algorithm="fricas")

[Out]

1/4*(8*x^6 - 128*x^2 - 32*(x^4 + 8*sqrt(-x^4 + 16) - 32)*arctan((sqrt(-x^4 + 16)
- 4)/x^2) - (x^6 - 32*x^2)*sqrt(-x^4 + 16))/(x^4 + 8*sqrt(-x^4 + 16) - 32)

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Sympy [A]  time = 6.10564, size = 80, normalized size = 2.76 \[ \begin{cases} - \frac{i x^{6}}{4 \sqrt{x^{4} - 16}} + \frac{4 i x^{2}}{\sqrt{x^{4} - 16}} - 4 i \operatorname{acosh}{\left (\frac{x^{2}}{4} \right )} & \text{for}\: \frac{\left |{x^{4}}\right |}{16} > 1 \\\frac{x^{6}}{4 \sqrt{- x^{4} + 16}} - \frac{4 x^{2}}{\sqrt{- x^{4} + 16}} + 4 \operatorname{asin}{\left (\frac{x^{2}}{4} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-I*x**6/(4*sqrt(x**4 - 16)) + 4*I*x**2/sqrt(x**4 - 16) - 4*I*acosh(x*
*2/4), Abs(x**4)/16 > 1), (x**6/(4*sqrt(-x**4 + 16)) - 4*x**2/sqrt(-x**4 + 16) +
4*asin(x**2/4), True))

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GIAC/XCAS [A]  time = 0.220729, size = 31, normalized size = 1.07 \[ -\frac{1}{4} \, \sqrt{-x^{4} + 16} x^{2} + 4 \, \arcsin \left (\frac{1}{4} \, x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 16),x, algorithm="giac")

[Out]

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