3.1.41 \(\int x^5 \sqrt {-1+x^6} \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{9} \left (x^6-1\right )^{3/2} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[-1 + x^6],x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[-1 + x^6],x]

[Out]

(-1 + x^6)^(3/2)/9

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (-1+x^6\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*Sqrt[-1 + x^6],x]

[Out]

(-1 + x^6)^(3/2)/9

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fricas [A]  time = 0.44, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

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giac [A]  time = 0.51, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

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maple [A]  time = 0.08, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9}\)	\(10\)
default	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9}\)	\(10\)
risch	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9}\)	\(10\)
trager	\(\left (\frac {x^{6}}{9}-\frac {1}{9}\right ) \sqrt {x^{6}-1}\)	\(16\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \sqrt {x^{6}-1}}{9}\)	\(30\)
meijerg	\(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (-2 x^{6}+2\right ) \sqrt {-x^{6}+1}}{3}\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

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mupad [B]  time = 0.19, size = 9, normalized size = 0.69 \begin {gather*} \frac {{\left (x^6-1\right )}^{3/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^6 - 1)^(1/2),x)

[Out]

(x^6 - 1)^(3/2)/9

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sympy [B]  time = 0.27, size = 22, normalized size = 1.69 \begin {gather*} \frac {x^{6} \sqrt {x^{6} - 1}}{9} - \frac {\sqrt {x^{6} - 1}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(x**6-1)**(1/2),x)

[Out]

x**6*sqrt(x**6 - 1)/9 - sqrt(x**6 - 1)/9

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