3.1518 \(\int \frac{1}{x^5 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{x^8+1}}{4 x^4} \]

[Out]

-Sqrt[1 + x^8]/(4*x^4)

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Rubi [A]  time = 0.0146453, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{x^8+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(4*x^4)

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Rubi in Sympy [A]  time = 2.27655, size = 14, normalized size = 0.88 \[ - \frac{\sqrt{x^{8} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8+1)**(1/2),x)

[Out]

-sqrt(x**8 + 1)/(4*x**4)

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Mathematica [A]  time = 0.00890641, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^8+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(4*x^4)

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Maple [A]  time = 0.005, size = 13, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44059, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^{8} + 1}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^5),x, algorithm="maxima")

[Out]

-1/4*sqrt(x^8 + 1)/x^4

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Fricas [A]  time = 0.232099, size = 27, normalized size = 1.69 \[ \frac{1}{4 \,{\left (x^{8} - \sqrt{x^{8} + 1} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^5),x, algorithm="fricas")

[Out]

1/4/(x^8 - sqrt(x^8 + 1)*x^4)

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Sympy [A]  time = 2.18693, size = 12, normalized size = 0.75 \[ - \frac{\sqrt{1 + \frac{1}{x^{8}}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8+1)**(1/2),x)

[Out]

-sqrt(1 + x**(-8))/4

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GIAC/XCAS [A]  time = 0.226487, size = 12, normalized size = 0.75 \[ -\frac{1}{4} \, \sqrt{\frac{1}{x^{8}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^5),x, algorithm="giac")

[Out]

-1/4*sqrt(1/x^8 + 1)