3.494 \(\int \frac {x^5}{\sqrt {-1-x^3}} \, dx\)

Optimal. Leaf size=31 \[ \frac {2}{9} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, 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} \[ \frac {2}{9} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[-1 - x^3],x]

[Out]

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

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^5}{\sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.65 \[ -\frac {2}{9} \sqrt {-x^3-1} \left (x^3-2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3]*(-2 + x^3))/9

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fricas [A]  time = 0.73, size = 16, normalized size = 0.52 \[ -\frac {2}{9} \, {\left (x^{3} - 2\right )} \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(x^3 - 2)*sqrt(-x^3 - 1)

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giac [A]  time = 0.15, size = 23, normalized size = 0.74 \[ \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 28, normalized size = 0.90 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (x^{3}-2\right )}{9 \sqrt {-x^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 23, normalized size = 0.74 \[ \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 16, normalized size = 0.52 \[ -\frac {2\,\sqrt {-x^3-1}\,\left (x^3-2\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(- x^3 - 1)^(1/2),x)

[Out]

-(2*(- x^3 - 1)^(1/2)*(x^3 - 2))/9

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sympy [A]  time = 0.66, size = 29, normalized size = 0.94 \[ - \frac {2 x^{3} \sqrt {- x^{3} - 1}}{9} + \frac {4 \sqrt {- x^{3} - 1}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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