3.1381 \(\int x^5 \sqrt{a^6-x^6} \, dx\)

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 = 0.003434, 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, Rules used = {261} \[ -\frac{1}{9} \left (a^6-x^6\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a^6 - x^6],x]

[Out]

-(a^6 - x^6)^(3/2)/9

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^5 \sqrt{a^6-x^6} \, dx &=-\frac{1}{9} \left (a^6-x^6\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0047003, 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 = 0.006, size = 43, normalized size = 2.5 \begin{align*} -{\frac{ \left ( a-x \right ) \left ( a+x \right ) \left ({a}^{2}+ax+{x}^{2} \right ) \left ({a}^{2}-ax+{x}^{2} \right ) }{9}\sqrt{{a}^{6}-{x}^{6}}} \end{align*}

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)*(a+x)*(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 = 0.993456, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a^6-x^6)^(1/2),x, algorithm="maxima")

[Out]

-1/9*(a^6 - x^6)^(3/2)

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Fricas [A]  time = 1.41476, size = 32, normalized size = 1.88 \begin{align*} -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a^6-x^6)^(1/2),x, algorithm="fricas")

[Out]

-1/9*(a^6 - x^6)^(3/2)

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Sympy [B]  time = 0.323603, size = 29, normalized size = 1.71 \begin{align*} - \frac{a^{6} \sqrt{a^{6} - x^{6}}}{9} + \frac{x^{6} \sqrt{a^{6} - x^{6}}}{9} \end{align*}

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 [A]  time = 1.17214, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{9} \,{\left (a^{6} - x^{6}\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a^6-x^6)^(1/2),x, algorithm="giac")

[Out]

-1/9*(a^6 - x^6)^(3/2)