3.14.81 \(\int x^5 \sqrt {a^6-x^6} \, dx\) [1381]

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

[Out]

-1/9*(a^6-x^6)^(3/2)

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {267} \begin {gather*} -\frac {1}{9} \left (a^6-x^6\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a^6 - x^6],x]

[Out]

-1/9*(a^6 - x^6)^(3/2)

Rule 267

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*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{9} \left (a^6-x^6\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a^6 - x^6],x]

[Out]

-1/9*(a^6 - x^6)^(3/2)

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Maple [A]
time = 0.17, size = 14, normalized size = 0.82

method	result	size
derivativedivides	\(-\frac {\left (a^{6}-x^{6}\right )^{\frac {3}{2}}}{9}\)	\(14\)
default	\(-\frac {\left (a^{6}-x^{6}\right )^{\frac {3}{2}}}{9}\)	\(14\)
risch	\(-\frac {\left (a^{6}-x^{6}\right )^{\frac {3}{2}}}{9}\)	\(14\)
trager	\(\left (-\frac {a^{6}}{9}+\frac {x^{6}}{9}\right ) \sqrt {a^{6}-x^{6}}\)	\(24\)
gosper	\(-\frac {\left (a -x \right ) \left (x +a \right ) \left (a^{2}+a x +x^{2}\right ) \left (a^{2}-a x +x^{2}\right ) \sqrt {a^{6}-x^{6}}}{9}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a^6-x^6)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/9*(a^6-x^6)^(3/2)

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Maxima [A]
time = 0.30, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{9} \, {\left (a^{6} - x^{6}\right )}^{\frac {3}{2}} \end {gather*}

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 = 0.35, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{9} \, {\left (a^{6} - x^{6}\right )}^{\frac {3}{2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 0.09, size = 29, normalized size = 1.71 \begin {gather*} - \frac {a^{6} \sqrt {a^{6} - x^{6}}}{9} + \frac {x^{6} \sqrt {a^{6} - x^{6}}}{9} \end {gather*}

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 = 1.49, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{9} \, {\left (a^{6} - x^{6}\right )}^{\frac {3}{2}} \end {gather*}

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)

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Mupad [B]
time = 1.23, size = 13, normalized size = 0.76 \begin {gather*} -\frac {{\left (a^6-x^6\right )}^{3/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a^6 - x^6)^(1/2),x)

[Out]

-(a^6 - x^6)^(3/2)/9

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