3.493 \(\int \frac {x^8}{\sqrt {-1-x^3}} \, dx\)

Optimal. Leaf size=46 \[ -\frac {2}{15} \left (-x^3-1\right )^{5/2}-\frac {4}{9} \left (-x^3-1\right )^{3/2}-\frac {2}{3} \sqrt {-x^3-1} \]

[Out]

-4/9*(-x^3-1)^(3/2)-2/15*(-x^3-1)^(5/2)-2/3*(-x^3-1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2}{15} \left (-x^3-1\right )^{5/2}-\frac {4}{9} \left (-x^3-1\right )^{3/2}-\frac {2}{3} \sqrt {-x^3-1} \]

Antiderivative was successfully verified.

[In]

Int[x^8/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3 - (4*(-1 - x^3)^(3/2))/9 - (2*(-1 - x^3)^(5/2))/15

Rule 43

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 266

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*} \int \frac {x^8}{\sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-1-x}}+2 \sqrt {-1-x}+(-1-x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {-1-x^3}-\frac {4}{9} \left (-1-x^3\right )^{3/2}-\frac {2}{15} \left (-1-x^3\right )^{5/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 27, normalized size = 0.59 \[ -\frac {2}{45} \sqrt {-x^3-1} \left (3 x^6-4 x^3+8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3]*(8 - 4*x^3 + 3*x^6))/45

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 23, normalized size = 0.50 \[ -\frac {2}{45} \, {\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3-1)^(1/2),x, algorithm="fricas")

[Out]

-2/45*(3*x^6 - 4*x^3 + 8)*sqrt(-x^3 - 1)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 41, normalized size = 0.89 \[ -\frac {2}{15} \, {\left (x^{3} + 1\right )}^{2} \sqrt {-x^{3} - 1} - \frac {4}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3-1)^(1/2),x, algorithm="giac")

[Out]

-2/15*(x^3 + 1)^2*sqrt(-x^3 - 1) - 4/9*(-x^3 - 1)^(3/2) - 2/3*sqrt(-x^3 - 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 35, normalized size = 0.76 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (3 x^{6}-4 x^{3}+8\right )}{45 \sqrt {-x^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(-x^3-1)^(1/2),x)

[Out]

2/45*(x+1)*(x^2-x+1)*(3*x^6-4*x^3+8)/(-x^3-1)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 34, normalized size = 0.74 \[ -\frac {2}{15} \, {\left (-x^{3} - 1\right )}^{\frac {5}{2}} - \frac {4}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3-1)^(1/2),x, algorithm="maxima")

[Out]

-2/15*(-x^3 - 1)^(5/2) - 4/9*(-x^3 - 1)^(3/2) - 2/3*sqrt(-x^3 - 1)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 23, normalized size = 0.50 \[ -\frac {2\,\sqrt {-x^3-1}\,\left (3\,x^6-4\,x^3+8\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(- x^3 - 1)^(1/2),x)

[Out]

-(2*(- x^3 - 1)^(1/2)*(3*x^6 - 4*x^3 + 8))/45

________________________________________________________________________________________

sympy [A]  time = 0.81, size = 46, normalized size = 1.00 \[ - \frac {2 x^{6} \sqrt {- x^{3} - 1}}{15} + \frac {8 x^{3} \sqrt {- x^{3} - 1}}{45} - \frac {16 \sqrt {- x^{3} - 1}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-x**3-1)**(1/2),x)

[Out]

-2*x**6*sqrt(-x**3 - 1)/15 + 8*x**3*sqrt(-x**3 - 1)/45 - 16*sqrt(-x**3 - 1)/45

________________________________________________________________________________________