3.2293 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

[Out]

-((a + b*x^(1/3))^3/(a*x))

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Rubi [A]  time = 0.0162551, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2/x^2,x]

[Out]

-((a + b*x^(1/3))^3/(a*x))

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Rubi in Sympy [A]  time = 2.85974, size = 14, normalized size = 0.74 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{3}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**(1/3))**3/(a*x)

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Mathematica [A]  time = 0.0149112, size = 28, normalized size = 1.47 \[ -\frac{a^2}{x}-\frac{3 a b}{x^{2/3}}-\frac{3 b^2}{\sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 25, normalized size = 1.3 \[ -{\frac{{a}^{2}}{x}}-3\,{\frac{ab}{{x}^{2/3}}}-3\,{\frac{{b}^{2}}{\sqrt [3]{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.47163, size = 32, normalized size = 1.68 \[ -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21334, size = 32, normalized size = 1.68 \[ -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.14483, size = 26, normalized size = 1.37 \[ - \frac{a^{2}}{x} - \frac{3 a b}{x^{\frac{2}{3}}} - \frac{3 b^{2}}{\sqrt [3]{x}} \]

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/x**(2/3) - 3*b**2/x**(1/3)

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GIAC/XCAS [A]  time = 0.260883, size = 32, normalized size = 1.68 \[ -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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