3.916 \(\int \frac{x^7}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\frac{\sqrt{x^4+1}}{2} \]

[Out]

-Sqrt[1 + x^4]/2 + (1 + x^4)^(3/2)/6

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Rubi [A]  time = 0.0316514, antiderivative size = 27, 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 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\frac{\sqrt{x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[1 + x^4],x]

[Out]

-Sqrt[1 + x^4]/2 + (1 + x^4)^(3/2)/6

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Rubi in Sympy [A]  time = 3.41998, size = 19, normalized size = 0.7 \[ \frac{\left (x^{4} + 1\right )^{\frac{3}{2}}}{6} - \frac{\sqrt{x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**4+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00923823, size = 18, normalized size = 0.67 \[ \frac{1}{6} \left (x^4-2\right ) \sqrt{x^4+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/Sqrt[1 + x^4],x]

[Out]

((-2 + x^4)*Sqrt[1 + x^4])/6

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Maple [A]  time = 0.007, size = 15, normalized size = 0.6 \[{\frac{{x}^{4}-2}{6}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43838, size = 26, normalized size = 0.96 \[ \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261307, size = 19, normalized size = 0.7 \[ \frac{1}{6} \, \sqrt{x^{4} + 1}{\left (x^{4} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

1/6*sqrt(x^4 + 1)*(x^4 - 2)

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Sympy [A]  time = 1.27964, size = 22, normalized size = 0.81 \[ \frac{x^{4} \sqrt{x^{4} + 1}}{6} - \frac{\sqrt{x^{4} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**4+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210199, size = 26, normalized size = 0.96 \[ \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

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