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]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.64, size = 26, normalized size = 2.00 \[ -\frac {3 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \left (\frac {x^{2} + 1}{x^{2}}\right )^{\frac {2}{3}}}{16 \, x^{4}} \]

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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giac [B]  time = 0.17, size = 23, normalized size = 1.77 \[ -\frac {3 \, {\left (x^{2} + 1\right )}^{2} \left (\frac {x^{2} + 1}{x^{2}}\right )^{\frac {2}{3}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 22, normalized size = 1.69 \[ -\frac {3 \left (x^{2}+1\right ) \left (\frac {x^{2}+1}{x^{2}}\right )^{\frac {5}{3}}}{16 x^{2}} \]

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 = 0.85, size = 9, normalized size = 0.69 \[ -\frac {3}{16} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {8}{3}} \]

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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mupad [B]  time = 1.42, size = 19, normalized size = 1.46 \[ -\frac {3\,{\left (\frac {1}{x^2}+1\right )}^{2/3}\,{\left (x^2+1\right )}^2}{16\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x^2 + 1)^(5/3)/x^3,x)

[Out]

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

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sympy [B]  time = 2.29, size = 48, normalized size = 3.69 \[ - \frac {3 \left (1 + \frac {1}{x^{2}}\right )^{\frac {2}{3}}}{16} - \frac {3 \left (1 + \frac {1}{x^{2}}\right )^{\frac {2}{3}}}{8 x^{2}} - \frac {3 \left (1 + \frac {1}{x^{2}}\right )^{\frac {2}{3}}}{16 x^{4}} \]

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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