3.936 \(\int \frac{x^7}{\left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

1/(2*Sqrt[1 + x^4]) + Sqrt[1 + x^4]/2

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Rubi [A]  time = 0.0318335, 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 \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 + x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 + x^4]) + Sqrt[1 + x^4]/2

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Rubi in Sympy [A]  time = 3.32849, size = 20, normalized size = 0.74 \[ \frac{\sqrt{x^{4} + 1}}{2} + \frac{1}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**4+1)**(3/2),x)

[Out]

sqrt(x**4 + 1)/2 + 1/(2*sqrt(x**4 + 1))

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Mathematica [A]  time = 0.00999915, size = 18, normalized size = 0.67 \[ \frac{x^4+2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 + x^4)^(3/2),x]

[Out]

(2 + x^4)/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.006, size = 15, normalized size = 0.6 \[{\frac{{x}^{4}+2}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44417, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{x^{4} + 1} + \frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)

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Fricas [A]  time = 0.248308, size = 19, normalized size = 0.7 \[ \frac{x^{4} + 2}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x^4 + 2)/sqrt(x^4 + 1)

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Sympy [A]  time = 2.33139, size = 22, normalized size = 0.81 \[ \frac{x^{4}}{2 \sqrt{x^{4} + 1}} + \frac{1}{\sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**4+1)**(3/2),x)

[Out]

x**4/(2*sqrt(x**4 + 1)) + 1/sqrt(x**4 + 1)

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GIAC/XCAS [A]  time = 0.220836, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{x^{4} + 1} + \frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)