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]

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.85, size = 21, normalized size = 0.52 \[ \frac {1}{30} \, {\left (3 \, x^{8} - 4 \, x^{4} + 8\right )} \sqrt {x^{4} + 1} \]

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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giac [A]  time = 0.15, size = 28, normalized size = 0.70 \[ \frac {1}{10} \, {\left (x^{4} + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x^{4} + 1} \]

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)

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maple [A]  time = 0.00, size = 22, normalized size = 0.55 \[ \frac {\sqrt {x^{4}+1}\, \left (3 x^{8}-4 x^{4}+8\right )}{30} \]

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 = 1.35, size = 28, normalized size = 0.70 \[ \frac {1}{10} \, {\left (x^{4} + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x^{4} + 1} \]

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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mupad [B]  time = 1.16, size = 20, normalized size = 0.50 \[ \sqrt {x^4+1}\,\left (\frac {x^8}{10}-\frac {2\,x^4}{15}+\frac {4}{15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^4 + 1)^(1/2),x)

[Out]

(x^4 + 1)^(1/2)*(x^8/10 - (2*x^4)/15 + 4/15)

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sympy [A]  time = 2.19, size = 39, normalized size = 0.98 \[ \frac {x^{8} \sqrt {x^{4} + 1}}{10} - \frac {2 x^{4} \sqrt {x^{4} + 1}}{15} + \frac {4 \sqrt {x^{4} + 1}}{15} \]

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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