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

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

[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 = 0.0186933, 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} \[ -\frac{2}{15} \left (1-x^3\right )^{5/2}+\frac{4}{9} \left (1-x^3\right )^{3/2}-\frac{2 \sqrt{1-x^3}}{3} \]

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 = 0.0086716, size = 27, normalized size = 0.59 \[ -\frac{2}{45} \sqrt{1-x^3} \left (3 x^6+4 x^3+8\right ) \]

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 = 0.004, size = 33, normalized size = 0.7 \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 = 0.999418, 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 = 1.4613, 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 = 0.897993, size = 42, normalized size = 0.91 \begin{align*} - \frac{2 x^{6} \sqrt{1 - x^{3}}}{15} - \frac{8 x^{3} \sqrt{1 - x^{3}}}{45} - \frac{16 \sqrt{1 - x^{3}}}{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(1 - x**3)/15 - 8*x**3*sqrt(1 - x**3)/45 - 16*sqrt(1 - x**3)/45

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Giac [A]  time = 1.1423, 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)