3.2393 \(\int \frac{a+\frac{b}{\sqrt [3]{x}}}{x^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Rubi [A]  time = 0.0161339, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))/x^2,x]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Rubi in Sympy [A]  time = 2.90105, size = 14, normalized size = 0.82 \[ - \frac{a}{x} - \frac{3 b}{4 x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x - 3*b/(4*x**(4/3))

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Mathematica [A]  time = 0.0089736, size = 17, normalized size = 1. \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Maple [A]  time = 0.008, size = 14, normalized size = 0.8 \[ -{\frac{3\,b}{4}{x}^{-{\frac{4}{3}}}}-{\frac{a}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.45291, size = 63, normalized size = 3.71 \[ -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4}}{4 \, b^{3}} + \frac{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{2}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222576, size = 20, normalized size = 1.18 \[ -\frac{4 \, a x^{\frac{1}{3}} + 3 \, b}{4 \, x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*a*x^(1/3) + 3*b)/x^(4/3)

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Sympy [A]  time = 2.4667, size = 14, normalized size = 0.82 \[ - \frac{a}{x} - \frac{3 b}{4 x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x - 3*b/(4*x**(4/3))

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GIAC/XCAS [A]  time = 0.210327, size = 20, normalized size = 1.18 \[ -\frac{4 \, a x^{\frac{1}{3}} + 3 \, b}{4 \, x^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*a*x^(1/3) + 3*b)/x^(4/3)