3.958 \(\int \frac{x^7}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt{16-x^4} \]

[Out]

-8*Sqrt[16 - x^4] + (16 - x^4)^(3/2)/6

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Rubi [A]  time = 0.0404775, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[16 - x^4],x]

[Out]

-8*Sqrt[16 - x^4] + (16 - x^4)^(3/2)/6

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Rubi in Sympy [A]  time = 4.79858, size = 19, normalized size = 0.66 \[ \frac{\left (- x^{4} + 16\right )^{\frac{3}{2}}}{6} - 8 \sqrt{- x^{4} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-x**4+16)**(1/2),x)

[Out]

(-x**4 + 16)**(3/2)/6 - 8*sqrt(-x**4 + 16)

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Mathematica [A]  time = 0.0128521, size = 20, normalized size = 0.69 \[ -\frac{1}{6} \sqrt{16-x^4} \left (x^4+32\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/Sqrt[16 - x^4],x]

[Out]

-(Sqrt[16 - x^4]*(32 + x^4))/6

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Maple [A]  time = 0.007, size = 28, normalized size = 1. \[{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) \left ({x}^{4}+32 \right ) }{6}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-x^4+16)^(1/2),x)

[Out]

1/6*(-2+x)*(2+x)*(x^2+4)*(x^4+32)/(-x^4+16)^(1/2)

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Maxima [A]  time = 1.43153, size = 31, normalized size = 1.07 \[ \frac{1}{6} \,{\left (-x^{4} + 16\right )}^{\frac{3}{2}} - 8 \, \sqrt{-x^{4} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 16),x, algorithm="maxima")

[Out]

1/6*(-x^4 + 16)^(3/2) - 8*sqrt(-x^4 + 16)

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Fricas [A]  time = 0.262264, size = 22, normalized size = 0.76 \[ -\frac{1}{6} \,{\left (x^{4} + 32\right )} \sqrt{-x^{4} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 16),x, algorithm="fricas")

[Out]

-1/6*(x^4 + 32)*sqrt(-x^4 + 16)

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Sympy [A]  time = 1.28591, size = 26, normalized size = 0.9 \[ - \frac{x^{4} \sqrt{- x^{4} + 16}}{6} - \frac{16 \sqrt{- x^{4} + 16}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-x**4+16)**(1/2),x)

[Out]

-x**4*sqrt(-x**4 + 16)/6 - 16*sqrt(-x**4 + 16)/3

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GIAC/XCAS [A]  time = 0.21355, size = 31, normalized size = 1.07 \[ \frac{1}{6} \,{\left (-x^{4} + 16\right )}^{\frac{3}{2}} - 8 \, \sqrt{-x^{4} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 16),x, algorithm="giac")

[Out]

1/6*(-x^4 + 16)^(3/2) - 8*sqrt(-x^4 + 16)