3.874 \(\int \frac{x^{11}}{\sqrt{1-x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0201704, antiderivative size = 46, normalized size of antiderivative = 1., 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} \[ -\frac{1}{10} \left (1-x^4\right )^{5/2}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[1 - x^4],x]

[Out]

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

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]]

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])

Rubi steps

\begin{align*} \int \frac{x^{11}}{\sqrt{1-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.0087716, size = 27, normalized size = 0.59 \[ -\frac{1}{30} \sqrt{1-x^4} \left (3 x^8+4 x^4+8\right ) \]

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 = 0.005, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 3\,{x}^{8}+4\,{x}^{4}+8 \right ) }{30}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.997233, size = 46, normalized size = 1. \begin{align*} -\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{align*}

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 = 1.45496, size = 57, normalized size = 1.24 \begin{align*} -\frac{1}{30} \,{\left (3 \, x^{8} + 4 \, x^{4} + 8\right )} \sqrt{-x^{4} + 1} \end{align*}

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 = 1.69958, size = 41, normalized size = 0.89 \begin{align*} - \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{align*}

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 = 1.12535, size = 55, normalized size = 1.2 \begin{align*} -\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{align*}

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)