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

Optimal. Leaf size=38 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.0137624, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]

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 = 0.0078431, size = 23, normalized size = 0.61 \[ \frac{x^8-4 x^4-8}{6 \sqrt{x^4+1}} \]

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 = 0.074, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{8}-4\,{x}^{4}-8}{6}{\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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.993598, size = 38, normalized size = 1. \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 = 1.43311, size = 50, normalized size = 1.32 \begin{align*} \frac{x^{8} - 4 \, x^{4} - 8}{6 \, \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="fricas")

[Out]

1/6*(x^8 - 4*x^4 - 8)/sqrt(x^4 + 1)

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Sympy [A]  time = 2.22573, size = 39, normalized size = 1.03 \begin{align*} \frac{x^{8}}{6 \sqrt{x^{4} + 1}} - \frac{2 x^{4}}{3 \sqrt{x^{4} + 1}} - \frac{4}{3 \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)

[Out]

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

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Giac [A]  time = 1.15364, size = 38, normalized size = 1. \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)