3.1379 \(\int x^5 \sqrt{a^6-x^6} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{9} \left (a^6-x^6\right )^{3/2} \]

[Out]

-(a^6 - x^6)^(3/2)/9

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Rubi [A]  time = 0.0089976, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{1}{9} \left (a^6-x^6\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a^6 - x^6],x]

[Out]

-(a^6 - x^6)^(3/2)/9

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Rubi in Sympy [A]  time = 2.10305, size = 12, normalized size = 0.71 \[ - \frac{\left (a^{6} - x^{6}\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(a**6-x**6)**(1/2),x)

[Out]

-(a**6 - x**6)**(3/2)/9

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Mathematica [A]  time = 0.00959021, size = 17, normalized size = 1. \[ -\frac{1}{9} \left (a^6-x^6\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a^6 - x^6],x]

[Out]

-(a^6 - x^6)^(3/2)/9

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Maple [B]  time = 0.01, size = 43, normalized size = 2.5 \[ -{\frac{ \left ( a-x \right ) \left ( x+a \right ) \left ({a}^{2}+ax+{x}^{2} \right ) \left ({a}^{2}-ax+{x}^{2} \right ) }{9}\sqrt{{a}^{6}-{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(a^6-x^6)^(1/2),x)

[Out]

-1/9*(a-x)*(x+a)*(a^2+a*x+x^2)*(a^2-a*x+x^2)*(a^6-x^6)^(1/2)

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Maxima [A]  time = 1.43565, size = 18, normalized size = 1.06 \[ -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a^6 - x^6)*x^5,x, algorithm="maxima")

[Out]

-1/9*(a^6 - x^6)^(3/2)

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Fricas [A]  time = 0.220685, size = 18, normalized size = 1.06 \[ -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a^6 - x^6)*x^5,x, algorithm="fricas")

[Out]

-1/9*(a^6 - x^6)^(3/2)

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Sympy [A]  time = 0.888946, size = 29, normalized size = 1.71 \[ - \frac{a^{6} \sqrt{a^{6} - x^{6}}}{9} + \frac{x^{6} \sqrt{a^{6} - x^{6}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(a**6-x**6)**(1/2),x)

[Out]

-a**6*sqrt(a**6 - x**6)/9 + x**6*sqrt(a**6 - x**6)/9

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GIAC/XCAS [A]  time = 0.221193, size = 18, normalized size = 1.06 \[ -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a^6 - x^6)*x^5,x, algorithm="giac")

[Out]

-1/9*(a^6 - x^6)^(3/2)