\(\int x^5 \sqrt {-1+x^6} \, dx\) [41]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \left (-1+x^6\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \left (x^6-1\right )^{3/2} \]

[In]

Int[x^5*Sqrt[-1 + x^6],x]

[Out]

(-1 + x^6)^(3/2)/9

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \left (-1+x^6\right )^{3/2} \]

[In]

Integrate[x^5*Sqrt[-1 + x^6],x]

[Out]

(-1 + x^6)^(3/2)/9

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 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	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9}\)	\(10\)
pseudoelliptic	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9}\)	\(10\)
trager	\(\left (\frac {x^{6}}{9}-\frac {1}{9}\right ) \sqrt {x^{6}-1}\)	\(16\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \sqrt {x^{6}-1}}{9}\)	\(30\)
meijerg	\(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (-2 x^{6}+2\right ) \sqrt {-x^{6}+1}}{3}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {x^{6} \sqrt {x^{6} - 1}}{9} - \frac {\sqrt {x^{6} - 1}}{9} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 5.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^5 \sqrt {-1+x^6} \, dx=\frac {{\left (x^6-1\right )}^{3/2}}{9} \]

[In]

int(x^5*(x^6 - 1)^(1/2),x)

[Out]

(x^6 - 1)^(3/2)/9