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

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 = 0.0412065, 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 \[ \frac{1}{10} \left (x^4+1\right )^{5/2}-\frac{1}{3} \left (x^4+1\right )^{3/2}+\frac{\sqrt{x^4+1}}{2} \]

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

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Rubi in Sympy [A]  time = 4.11713, size = 29, normalized size = 0.72 \[ \frac{\left (x^{4} + 1\right )^{\frac{5}{2}}}{10} - \frac{\left (x^{4} + 1\right )^{\frac{3}{2}}}{3} + \frac{\sqrt{x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**4+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0147247, 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 = 0.006, size = 22, normalized size = 0.6 \[{\frac{3\,{x}^{8}-4\,{x}^{4}+8}{30}\sqrt{{x}^{4}+1}} \]

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 = 1.43604, size = 38, normalized size = 0.95 \[ \frac{1}{10} \,{\left (x^{4} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/sqrt(x^4 + 1),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 = 0.265958, size = 28, normalized size = 0.7 \[ \frac{1}{30} \,{\left (3 \, x^{8} - 4 \, x^{4} + 8\right )} \sqrt{x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.08776, 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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GIAC/XCAS [A]  time = 0.213109, size = 38, normalized size = 0.95 \[ \frac{1}{10} \,{\left (x^{4} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

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