3.1956 \(\int \frac{\left (1+\frac{1}{x^2}\right )^{5/3}}{x^3} \, dx\)

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 = 0.0168628, 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 \[ -\frac{3}{16} \left (\frac{1}{x^2}+1\right )^{8/3} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(-2))^(5/3)/x^3,x]

[Out]

(-3*(1 + x^(-2))^(8/3))/16

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Rubi in Sympy [A]  time = 1.67006, size = 14, normalized size = 1.08 \[ - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{8}{3}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+1/x**2)**(5/3)/x**3,x)

[Out]

-3*(1 + x**(-2))**(8/3)/16

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Mathematica [A]  time = 0.0200953, size = 23, normalized size = 1.77 \[ -\frac{3 \left (\frac{1}{x^2}+1\right )^{2/3} \left (x^2+1\right )^2}{16 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^(-2))^(5/3)/x^3,x]

[Out]

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

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Maple [B]  time = 0.004, size = 22, normalized size = 1.7 \[ -{\frac{3\,{x}^{2}+3}{16\,{x}^{2}} \left ({\frac{{x}^{2}+1}{{x}^{2}}} \right ) ^{{\frac{5}{3}}}} \]

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 = 1.43499, size = 12, normalized size = 0.92 \[ -\frac{3}{16} \,{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{8}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/16*(1/x^2 + 1)^(8/3)

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Fricas [A]  time = 0.242366, size = 35, normalized size = 2.69 \[ -\frac{3 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \left (\frac{x^{2} + 1}{x^{2}}\right )^{\frac{2}{3}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1/x^2 + 1)^(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 [A]  time = 8.54398, 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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{5}{3}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((1/x^2 + 1)^(5/3)/x^3, x)