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]

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

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Rubi [A]  time = 0.0185338, antiderivative size = 46, normalized size of antiderivative = 1., 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 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]]

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])

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 = 0.0077563, 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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 3\,{x}^{6}-4\,{x}^{3}+8 \right ) }{45}{\frac{1}{\sqrt{-{x}^{3}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02993, size = 46, normalized size = 1. \begin{align*} -\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} \end{align*}

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)

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Fricas [A]  time = 1.63207, size = 57, normalized size = 1.24 \begin{align*} -\frac{2}{45} \,{\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt{-x^{3} - 1} \end{align*}

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)

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Sympy [A]  time = 0.824093, size = 46, normalized size = 1. \begin{align*} - \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} \end{align*}

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

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Giac [A]  time = 1.10191, size = 55, normalized size = 1.2 \begin{align*} -\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} \end{align*}

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)