3.9.74 \(\int \frac {x^{11}}{\sqrt {1-x^4}} \, dx\) [874]

Optimal. Leaf size=46 \[ -\frac {1}{2} \sqrt {1-x^4}+\frac {1}{3} \left (1-x^4\right )^{3/2}-\frac {1}{10} \left (1-x^4\right )^{5/2} \]

[Out]

1/3*(-x^4+1)^(3/2)-1/10*(-x^4+1)^(5/2)-1/2*(-x^4+1)^(1/2)

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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 {1}{10} \left (1-x^4\right )^{5/2}+\frac {1}{3} \left (1-x^4\right )^{3/2}-\frac {\sqrt {1-x^4}}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[1 - x^4],x]

[Out]

-1/2*Sqrt[1 - x^4] + (1 - x^4)^(3/2)/3 - (1 - x^4)^(5/2)/10

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^{11}}{\sqrt {1-x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-2 \sqrt {1-x}+(1-x)^{3/2}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2} \sqrt {1-x^4}+\frac {1}{3} \left (1-x^4\right )^{3/2}-\frac {1}{10} \left (1-x^4\right )^{5/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.59 \begin {gather*} \frac {1}{30} \sqrt {1-x^4} \left (-8-4 x^4-3 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11/Sqrt[1 - x^4],x]

[Out]

(Sqrt[1 - x^4]*(-8 - 4*x^4 - 3*x^8))/30

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Maple [A]
time = 0.15, size = 34, normalized size = 0.74

method	result	size
trager	\(\left (-\frac {1}{10} x^{8}-\frac {2}{15} x^{4}-\frac {4}{15}\right ) \sqrt {-x^{4}+1}\)	\(23\)
risch	\(\frac {\left (3 x^{8}+4 x^{4}+8\right ) \left (x^{4}-1\right )}{30 \sqrt {-x^{4}+1}}\)	\(29\)
default	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right ) \left (3 x^{8}+4 x^{4}+8\right )}{30 \sqrt {-x^{4}+1}}\)	\(34\)
elliptic	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right ) \left (3 x^{8}+4 x^{4}+8\right )}{30 \sqrt {-x^{4}+1}}\)	\(34\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (3 x^{8}+4 x^{4}+8\right )}{30 \sqrt {-x^{4}+1}}\)	\(35\)
meijerg	\(-\frac {-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{8}+8 x^{4}+16\right ) \sqrt {-x^{4}+1}}{15}}{4 \sqrt {\pi }}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/30*(x^2-1)*(x^2+1)*(3*x^8+4*x^4+8)/(-x^4+1)^(1/2)

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Maxima [A]
time = 0.30, size = 34, normalized size = 0.74 \begin {gather*} -\frac {1}{10} \, {\left (-x^{4} + 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/10*(-x^4 + 1)^(5/2) + 1/3*(-x^4 + 1)^(3/2) - 1/2*sqrt(-x^4 + 1)

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Fricas [A]
time = 0.36, size = 23, normalized size = 0.50 \begin {gather*} -\frac {1}{30} \, {\left (3 \, x^{8} + 4 \, x^{4} + 8\right )} \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/30*(3*x^8 + 4*x^4 + 8)*sqrt(-x^4 + 1)

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Sympy [A]
time = 0.21, size = 41, normalized size = 0.89 \begin {gather*} - \frac {x^{8} \sqrt {1 - x^{4}}}{10} - \frac {2 x^{4} \sqrt {1 - x^{4}}}{15} - \frac {4 \sqrt {1 - x^{4}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(-x**4+1)**(1/2),x)

[Out]

-x**8*sqrt(1 - x**4)/10 - 2*x**4*sqrt(1 - x**4)/15 - 4*sqrt(1 - x**4)/15

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Giac [A]
time = 1.37, size = 41, normalized size = 0.89 \begin {gather*} -\frac {1}{10} \, {\left (x^{4} - 1\right )}^{2} \sqrt {-x^{4} + 1} + \frac {1}{3} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

-1/10*(x^4 - 1)^2*sqrt(-x^4 + 1) + 1/3*(-x^4 + 1)^(3/2) - 1/2*sqrt(-x^4 + 1)

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Mupad [B]
time = 1.20, size = 23, normalized size = 0.50 \begin {gather*} -\sqrt {1-x^4}\,\left (\frac {x^8}{10}+\frac {2\,x^4}{15}+\frac {4}{15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(1 - x^4)^(1/2),x)

[Out]

-(1 - x^4)^(1/2)*((2*x^4)/15 + x^8/10 + 4/15)

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