3.894 \(\int \frac{x^{11}}{(1-x^4)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0184253, antiderivative size = 42, 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} \[ -\frac{1}{6} \left (1-x^4\right )^{3/2}+\sqrt{1-x^4}+\frac{1}{2 \sqrt{1-x^4}} \]

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 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}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.0092291, size = 27, normalized size = 0.64 \[ \frac{-x^8-4 x^4+8}{6 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(1 - x^4)^(3/2),x]

[Out]

(8 - 4*x^4 - x^8)/(6*Sqrt[1 - x^4])

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Maple [A]  time = 0.003, size = 33, normalized size = 0.8 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{8}+4\,{x}^{4}-8 \right ) }{6} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.997337, size = 43, normalized size = 1.02 \begin{align*} -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \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)^(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 = 1.50461, size = 65, normalized size = 1.55 \begin{align*} \frac{{\left (x^{8} + 4 \, x^{4} - 8\right )} \sqrt{-x^{4} + 1}}{6 \,{\left (x^{4} - 1\right )}} \end{align*}

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 = 1.86091, size = 39, normalized size = 0.93 \begin{align*} - \frac{x^{8}}{6 \sqrt{1 - x^{4}}} - \frac{2 x^{4}}{3 \sqrt{1 - x^{4}}} + \frac{4}{3 \sqrt{1 - x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1948, size = 43, normalized size = 1.02 \begin{align*} -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \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)^(3/2),x, algorithm="giac")

[Out]

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