3.2292 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0345169, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^2 \log (x)+6 a b \sqrt [3]{x}+\frac{3}{2} b^2 x^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2/x,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{2} \log{\left (\sqrt [3]{x} \right )} + 6 a b \sqrt [3]{x} + 3 b^{2} \int ^{\sqrt [3]{x}} x\, dx \]

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*Integral(x, (x, x**(1/3)))

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Mathematica [A]  time = 0.014204, size = 28, normalized size = 1. \[ a^2 \log (x)+\frac{3}{2} b \sqrt [3]{x} \left (4 a+b \sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 23, normalized size = 0.8 \[ 6\,ab\sqrt [3]{x}+{\frac{3\,{b}^{2}}{2}{x}^{{\frac{2}{3}}}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2/x,x)

[Out]

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

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Maxima [A]  time = 1.44329, size = 30, normalized size = 1.07 \[ a^{2} \log \left (x\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218895, size = 34, normalized size = 1.21 \[ 3 \, a^{2} \log \left (x^{\frac{1}{3}}\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.628671, size = 27, normalized size = 0.96 \[ a^{2} \log{\left (x \right )} + 6 a b \sqrt [3]{x} + \frac{3 b^{2} x^{\frac{2}{3}}}{2} \]

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*x**(2/3)/2

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GIAC/XCAS [A]  time = 0.250871, size = 31, normalized size = 1.11 \[ a^{2}{\rm ln}\left ({\left | x \right |}\right ) + \frac{3}{2} \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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