3.2401 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2}{x} \, dx\)

Optimal. Leaf size=28 \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]

[Out]

(-3*b^2)/(2*x^(2/3)) - (6*a*b)/x^(1/3) + a^2*Log[x]

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Rubi [A]  time = 0.0486285, antiderivative size = 28, 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 \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2/x,x]

[Out]

(-3*b^2)/(2*x^(2/3)) - (6*a*b)/x^(1/3) + a^2*Log[x]

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Rubi in Sympy [A]  time = 7.9464, size = 32, normalized size = 1.14 \[ 3 a^{2} \log{\left (\sqrt [3]{x} \right )} - \frac{6 a b}{\sqrt [3]{x}} - \frac{3 b^{2}}{2 x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*log(x**(1/3)) - 6*a*b/x**(1/3) - 3*b**2/(2*x**(2/3))

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Mathematica [A]  time = 0.0369235, size = 27, normalized size = 0.96 \[ a^2 \log (x)-\frac{3 b \left (4 a \sqrt [3]{x}+b\right )}{2 x^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 23, normalized size = 0.8 \[ -{\frac{3\,{b}^{2}}{2}{x}^{-{\frac{2}{3}}}}-6\,{\frac{ab}{\sqrt [3]{x}}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2/x,x)

[Out]

-3/2*b^2/x^(2/3)-6*a*b/x^(1/3)+a^2*ln(x)

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Maxima [A]  time = 1.44041, size = 30, normalized size = 1.07 \[ a^{2} \log \left (x\right ) - \frac{6 \, a b}{x^{\frac{1}{3}}} - \frac{3 \, b^{2}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*log(x) - 6*a*b/x^(1/3) - 3/2*b^2/x^(2/3)

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Fricas [A]  time = 0.22528, size = 41, normalized size = 1.46 \[ \frac{3 \,{\left (2 \, a^{2} x^{\frac{2}{3}} \log \left (x^{\frac{1}{3}}\right ) - 4 \, a b x^{\frac{1}{3}} - b^{2}\right )}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.96242, size = 27, normalized size = 0.96 \[ a^{2} \log{\left (x \right )} - \frac{6 a b}{\sqrt [3]{x}} - \frac{3 b^{2}}{2 x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(x) - 6*a*b/x**(1/3) - 3*b**2/(2*x**(2/3))

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GIAC/XCAS [A]  time = 0.212894, size = 32, normalized size = 1.14 \[ a^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + b^{2}\right )}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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