\(\int \frac {x^5}{\sqrt {-1+x^3}} \, dx\) [476]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{3} \sqrt {-1+x^3}+\frac {2}{9} \left (-1+x^3\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{9} \left (x^3-1\right )^{3/2}+\frac {2 \sqrt {x^3-1}}{3} \]

[In]

Int[x^5/Sqrt[-1 + x^3],x]

[Out]

(2*Sqrt[-1 + x^3])/3 + (2*(-1 + x^3)^(3/2))/9

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx,x,x^3\right ) \\ & = \frac {2}{3} \sqrt {-1+x^3}+\frac {2}{9} \left (-1+x^3\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{9} \sqrt {-1+x^3} \left (2+x^3\right ) \]

[In]

Integrate[x^5/Sqrt[-1 + x^3],x]

[Out]

(2*Sqrt[-1 + x^3]*(2 + x^3))/9

Maple [A] (verified)

Time = 3.95 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {2 \left (x^{3}+2\right ) \sqrt {x^{3}-1}}{9}\)	\(15\)
pseudoelliptic	\(\frac {2 \left (x^{3}+2\right ) \sqrt {x^{3}-1}}{9}\)	\(15\)
trager	\(\left (\frac {2 x^{3}}{9}+\frac {4}{9}\right ) \sqrt {x^{3}-1}\)	\(16\)
default	\(\frac {2 x^{3} \sqrt {x^{3}-1}}{9}+\frac {4 \sqrt {x^{3}-1}}{9}\)	\(23\)
elliptic	\(\frac {2 x^{3} \sqrt {x^{3}-1}}{9}+\frac {4 \sqrt {x^{3}-1}}{9}\)	\(23\)
gosper	\(\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{3}+2\right )}{9 \sqrt {x^{3}-1}}\)	\(24\)
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right ) \sqrt {-x^{3}+1}}{6}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\)	\(51\)

[In]

int(x^5/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{9} \, {\left (x^{3} + 2\right )} \sqrt {x^{3} - 1} \]

[In]

integrate(x^5/(x^3-1)^(1/2),x, algorithm="fricas")

[Out]

2/9*(x^3 + 2)*sqrt(x^3 - 1)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2 x^{3} \sqrt {x^{3} - 1}}{9} + \frac {4 \sqrt {x^{3} - 1}}{9} \]

[In]

integrate(x**5/(x**3-1)**(1/2),x)

[Out]

2*x**3*sqrt(x**3 - 1)/9 + 4*sqrt(x**3 - 1)/9

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{9} \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {x^{3} - 1} \]

[In]

integrate(x^5/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

2/9*(x^3 - 1)^(3/2) + 2/3*sqrt(x^3 - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2}{9} \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {x^{3} - 1} \]

[In]

integrate(x^5/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

2/9*(x^3 - 1)^(3/2) + 2/3*sqrt(x^3 - 1)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^5}{\sqrt {-1+x^3}} \, dx=\frac {2\,\sqrt {x^3-1}\,\left (x^3+2\right )}{9} \]

[In]

int(x^5/(x^3 - 1)^(1/2),x)

[Out]

(2*(x^3 - 1)^(1/2)*(x^3 + 2))/9