3.5.94 \(\int \frac {x^5}{\sqrt {-1-x^3}} \, dx\) [494]

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

[Out]

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

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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 = {272, 45} \begin {gather*} \frac {2}{9} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \end {gather*}

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 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*} \int \frac {x^5}{\sqrt {-1-x^3}} \, dx &=\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*}

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

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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Maple [A]
time = 0.13, size = 27, normalized size = 0.87

method	result	size
trager	\(\left (-\frac {2 x^{3}}{9}+\frac {4}{9}\right ) \sqrt {-x^{3}-1}\)	\(18\)
risch	\(\frac {2 \left (x^{3}-2\right ) \left (x^{3}+1\right )}{9 \sqrt {-x^{3}-1}}\)	\(22\)
default	\(-\frac {2 x^{3} \sqrt {-x^{3}-1}}{9}+\frac {4 \sqrt {-x^{3}-1}}{9}\)	\(27\)
elliptic	\(-\frac {2 x^{3} \sqrt {-x^{3}-1}}{9}+\frac {4 \sqrt {-x^{3}-1}}{9}\)	\(27\)
gosper	\(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (x^{3}-2\right )}{9 \sqrt {-x^{3}-1}}\)	\(28\)
meijerg	\(-\frac {i \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {x^{3}+1}}{6}\right )}{3 \sqrt {\pi }}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.74 \begin {gather*} \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \end {gather*}

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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Fricas [A]
time = 0.37, size = 16, normalized size = 0.52 \begin {gather*} -\frac {2}{9} \, {\left (x^{3} - 2\right )} \sqrt {-x^{3} - 1} \end {gather*}

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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Sympy [A]
time = 0.08, size = 29, normalized size = 0.94 \begin {gather*} - \frac {2 x^{3} \sqrt {- x^{3} - 1}}{9} + \frac {4 \sqrt {- x^{3} - 1}}{9} \end {gather*}

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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Giac [A]
time = 2.85, size = 23, normalized size = 0.74 \begin {gather*} \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \end {gather*}

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

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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