3.1958 \(\int \frac{(1+\frac{1}{x^2})^{5/3}}{x^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{3}{16} \left (\frac{1}{x^2}+1\right )^{8/3} \]

[Out]

(-3*(1 + x^(-2))^(8/3))/16

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Rubi [A]  time = 0.0035764, antiderivative size = 13, 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 = {261} \[ -\frac{3}{16} \left (\frac{1}{x^2}+1\right )^{8/3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(-2))^(5/3)/x^3,x]

[Out]

(-3*(1 + x^(-2))^(8/3))/16

Rule 261

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

Rubi steps

\begin{align*} \int \frac{\left (1+\frac{1}{x^2}\right )^{5/3}}{x^3} \, dx &=-\frac{3}{16} \left (1+\frac{1}{x^2}\right )^{8/3}\\ \end{align*}

Mathematica [A]  time = 0.0072235, size = 21, normalized size = 1.62 \[ -\frac{3 \left (\frac{1}{x^2}+1\right )^{5/3} \left (x^2+1\right )}{16 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(-2))^(5/3)/x^3,x]

[Out]

(-3*(1 + x^(-2))^(5/3)*(1 + x^2))/(16*x^2)

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Maple [B]  time = 0.003, size = 22, normalized size = 1.7 \begin{align*} -{\frac{3\,{x}^{2}+3}{16\,{x}^{2}} \left ({\frac{{x}^{2}+1}{{x}^{2}}} \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+1/x^2)^(5/3)/x^3,x)

[Out]

-3/16/x^2*(x^2+1)*((x^2+1)/x^2)^(5/3)

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Maxima [A]  time = 0.960308, size = 12, normalized size = 0.92 \begin{align*} -\frac{3}{16} \,{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+1/x^2)^(5/3)/x^3,x, algorithm="maxima")

[Out]

-3/16*(1/x^2 + 1)^(8/3)

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Fricas [B]  time = 1.52766, size = 69, normalized size = 5.31 \begin{align*} -\frac{3 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \left (\frac{x^{2} + 1}{x^{2}}\right )^{\frac{2}{3}}}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+1/x^2)^(5/3)/x^3,x, algorithm="fricas")

[Out]

-3/16*(x^4 + 2*x^2 + 1)*((x^2 + 1)/x^2)^(2/3)/x^4

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Sympy [B]  time = 2.08207, size = 48, normalized size = 3.69 \begin{align*} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{16} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{8 x^{2}} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{16 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+1/x**2)**(5/3)/x**3,x)

[Out]

-3*(1 + x**(-2))**(2/3)/16 - 3*(1 + x**(-2))**(2/3)/(8*x**2) - 3*(1 + x**(-2))**(2/3)/(16*x**4)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{5}{3}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+1/x^2)^(5/3)/x^3,x, algorithm="giac")

[Out]

integrate((1/x^2 + 1)^(5/3)/x^3, x)