3.2402 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2}{x^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]

[Out]

(-3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3)) - a^2/x

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Rubi [A]  time = 0.0480176, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2/x^2,x]

[Out]

(-3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3)) - a^2/x

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Rubi in Sympy [A]  time = 8.01112, size = 29, normalized size = 0.91 \[ - \frac{a^{2}}{x} - \frac{3 a b}{2 x^{\frac{4}{3}}} - \frac{3 b^{2}}{5 x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/x - 3*a*b/(2*x**(4/3)) - 3*b**2/(5*x**(5/3))

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Mathematica [A]  time = 0.0201804, size = 32, normalized size = 1. \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^2/x^2,x]

[Out]

(-3*b^2)/(5*x^(5/3)) - (3*a*b)/(2*x^(4/3)) - a^2/x

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Maple [A]  time = 0.009, size = 25, normalized size = 0.8 \[ -{\frac{3\,{b}^{2}}{5}{x}^{-{\frac{5}{3}}}}-{\frac{3\,ab}{2}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2/x^2,x)

[Out]

-3/5*b^2/x^(5/3)-3/2*a*b/x^(4/3)-a^2/x

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Maxima [A]  time = 1.43103, size = 63, normalized size = 1.97 \[ -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5}}{5 \, b^{3}} + \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a}{2 \, b^{3}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/5*(a + b/x^(1/3))^5/b^3 + 3/2*(a + b/x^(1/3))^4*a/b^3 - (a + b/x^(1/3))^3*a^2
/b^3

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Fricas [A]  time = 0.220105, size = 35, normalized size = 1.09 \[ -\frac{10 \, a^{2} x^{\frac{2}{3}} + 15 \, a b x^{\frac{1}{3}} + 6 \, b^{2}}{10 \, x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2/x^2,x, algorithm="fricas")

[Out]

-1/10*(10*a^2*x^(2/3) + 15*a*b*x^(1/3) + 6*b^2)/x^(5/3)

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Sympy [A]  time = 3.11919, size = 29, normalized size = 0.91 \[ - \frac{a^{2}}{x} - \frac{3 a b}{2 x^{\frac{4}{3}}} - \frac{3 b^{2}}{5 x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**2/x**2,x)

[Out]

-a**2/x - 3*a*b/(2*x**(4/3)) - 3*b**2/(5*x**(5/3))

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GIAC/XCAS [A]  time = 0.211076, size = 35, normalized size = 1.09 \[ -\frac{10 \, a^{2} x^{\frac{2}{3}} + 15 \, a b x^{\frac{1}{3}} + 6 \, b^{2}}{10 \, x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(10*a^2*x^(2/3) + 15*a*b*x^(1/3) + 6*b^2)/x^(5/3)