3.917 \(\int \frac{x^{11}}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{10} \left (x^4+1\right )^{5/2}-\frac{1}{3} \left (x^4+1\right )^{3/2}+\frac{\sqrt{x^4+1}}{2} \]

[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 = 0.0143273, antiderivative size = 40, 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} \[ \frac{1}{10} \left (x^4+1\right )^{5/2}-\frac{1}{3} \left (x^4+1\right )^{3/2}+\frac{\sqrt{x^4+1}}{2} \]

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{\sqrt{1+x^4}}{2}-\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 = 0.0069453, size = 25, normalized size = 0.62 \[ \frac{1}{30} \sqrt{x^4+1} \left (3 x^8-4 x^4+8\right ) \]

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 = 0.004, size = 22, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{8}-4\,{x}^{4}+8}{30}\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*(x^4+1)^(1/2)*(3*x^8-4*x^4+8)

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Maxima [A]  time = 0.987845, size = 38, normalized size = 0.95 \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 = 1.5019, size = 54, normalized size = 1.35 \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 = 1.27381, size = 39, normalized size = 0.98 \begin{align*} \frac{x^{8} \sqrt{x^{4} + 1}}{10} - \frac{2 x^{4} \sqrt{x^{4} + 1}}{15} + \frac{4 \sqrt{x^{4} + 1}}{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(x**4 + 1)/10 - 2*x**4*sqrt(x**4 + 1)/15 + 4*sqrt(x**4 + 1)/15

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Giac [A]  time = 1.18601, size = 38, normalized size = 0.95 \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="giac")

[Out]

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