3.9.94 \(\int \frac {x^{11}}{(1-x^4)^{3/2}} \, dx\) [894]

Optimal. Leaf size=42 \[ \frac {1}{2 \sqrt {1-x^4}}+\sqrt {1-x^4}-\frac {1}{6} \left (1-x^4\right )^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 42, 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}{6} \left (1-x^4\right )^{3/2}+\sqrt {1-x^4}+\frac {1}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4] - (1 - x^4)^(3/2)/6

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}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{(1-x)^{3/2}}-\frac {2}{\sqrt {1-x}}+\sqrt {1-x}\right ) \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1-x^4}}+\sqrt {1-x^4}-\frac {1}{6} \left (1-x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 0.60 \begin {gather*} -\frac {-8+4 x^4+x^8}{6 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11/(1 - x^4)^(3/2),x]

[Out]

-1/6*(-8 + 4*x^4 + x^8)/Sqrt[1 - x^4]

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Maple [A]
time = 0.16, size = 22, normalized size = 0.52

method	result	size
default	\(-\frac {x^{8}+4 x^{4}-8}{6 \sqrt {-x^{4}+1}}\)	\(22\)
risch	\(-\frac {x^{8}+4 x^{4}-8}{6 \sqrt {-x^{4}+1}}\)	\(22\)
elliptic	\(-\frac {x^{8}+4 x^{4}-8}{6 \sqrt {-x^{4}+1}}\)	\(22\)
trager	\(\frac {\left (x^{8}+4 x^{4}-8\right ) \sqrt {-x^{4}+1}}{6 x^{4}-6}\)	\(29\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{8}+4 x^{4}-8\right )}{6 \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(33\)
meijerg	\(-\frac {\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 x^{8}-8 x^{4}+16\right )}{6 \sqrt {-x^{4}+1}}}{2 \sqrt {\pi }}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 32, normalized size = 0.76 \begin {gather*} -\frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} + \sqrt {-x^{4} + 1} + \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 28, normalized size = 0.67 \begin {gather*} \frac {{\left (x^{8} + 4 \, x^{4} - 8\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/6*(x^8 + 4*x^4 - 8)*sqrt(-x^4 + 1)/(x^4 - 1)

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Sympy [A]
time = 0.32, size = 54, normalized size = 1.29 \begin {gather*} \frac {x^{8} \sqrt {1 - x^{4}}}{6 x^{4} - 6} + \frac {4 x^{4} \sqrt {1 - x^{4}}}{6 x^{4} - 6} - \frac {8 \sqrt {1 - x^{4}}}{6 x^{4} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(-x**4+1)**(3/2),x)

[Out]

x**8*sqrt(1 - x**4)/(6*x**4 - 6) + 4*x**4*sqrt(1 - x**4)/(6*x**4 - 6) - 8*sqrt(1 - x**4)/(6*x**4 - 6)

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Giac [A]
time = 1.54, size = 32, normalized size = 0.76 \begin {gather*} -\frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} + \sqrt {-x^{4} + 1} + \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)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.19, size = 25, normalized size = 0.60 \begin {gather*} -\frac {{\left (x^4-1\right )}^2+6\,x^4-9}{6\,\sqrt {1-x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(1 - x^4)^(3/2),x)

[Out]

-((x^4 - 1)^2 + 6*x^4 - 9)/(6*(1 - x^4)^(1/2))

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