3.1520 \(\int \frac{1}{x^5 \sqrt{1+x^8}} \, dx\)

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 = 0.0028306, 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, Rules used = {264} \[ -\frac{\sqrt{x^8+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(4*x^4)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \sqrt{1+x^8}} \, dx &=-\frac{\sqrt{1+x^8}}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0023713, 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 = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{8}+1}} \end{align*}

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 = 0.943061, size = 16, normalized size = 1. \begin{align*} -\frac{\sqrt{x^{8} + 1}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(x^8 + 1)/x^4

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Fricas [A]  time = 1.2289, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{4} + \sqrt{x^{8} + 1}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

-1/4*(x^4 + sqrt(x^8 + 1))/x^4

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Sympy [A]  time = 0.644225, size = 12, normalized size = 0.75 \begin{align*} - \frac{\sqrt{1 + \frac{1}{x^{8}}}}{4} \end{align*}

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 [A]  time = 1.18013, size = 12, normalized size = 0.75 \begin{align*} -\frac{1}{4} \, \sqrt{\frac{1}{x^{8}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(1/x^8 + 1)