3.5.57 \(\int \frac {x^8}{\sqrt {1-x^3}} \, dx\) [457]

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

[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 = {272, 45} \begin {gather*} -\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} \end {gather*}

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 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^8}{\sqrt {1-x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 = 0.02, size = 27, normalized size = 0.59 \begin {gather*} -\frac {2}{45} \sqrt {1-x^3} \left (8+4 x^3+3 x^6\right ) \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 41, normalized size = 0.89

method	result	size
trager	\(\left (-\frac {2}{15} x^{6}-\frac {8}{45} x^{3}-\frac {16}{45}\right ) \sqrt {-x^{3}+1}\)	\(23\)
risch	\(\frac {2 \left (3 x^{6}+4 x^{3}+8\right ) \left (x^{3}-1\right )}{45 \sqrt {-x^{3}+1}}\)	\(29\)
gosper	\(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (3 x^{6}+4 x^{3}+8\right )}{45 \sqrt {-x^{3}+1}}\)	\(33\)
meijerg	\(-\frac {-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{6}+8 x^{3}+16\right ) \sqrt {-x^{3}+1}}{15}}{3 \sqrt {\pi }}\)	\(38\)
default	\(-\frac {2 x^{6} \sqrt {-x^{3}+1}}{15}-\frac {8 x^{3} \sqrt {-x^{3}+1}}{45}-\frac {16 \sqrt {-x^{3}+1}}{45}\)	\(41\)
elliptic	\(-\frac {2 x^{6} \sqrt {-x^{3}+1}}{15}-\frac {8 x^{3} \sqrt {-x^{3}+1}}{45}-\frac {16 \sqrt {-x^{3}+1}}{45}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 34, normalized size = 0.74 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 23, normalized size = 0.50 \begin {gather*} -\frac {2}{45} \, {\left (3 \, x^{6} + 4 \, x^{3} + 8\right )} \sqrt {-x^{3} + 1} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 42, normalized size = 0.91 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 1.96, size = 41, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(1 - x^3)^(1/2),x)

[Out]

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

________________________________________________________________________________________