3.2302 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^3}{x^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a^3}{x}-\frac{9 a^2 b}{2 x^{2/3}}-\frac{9 a b^2}{\sqrt [3]{x}}+b^3 \log (x) \]

[Out]

-(a^3/x) - (9*a^2*b)/(2*x^(2/3)) - (9*a*b^2)/x^(1/3) + b^3*Log[x]

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Rubi [A]  time = 0.0543552, antiderivative size = 39, 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 \[ -\frac{a^3}{x}-\frac{9 a^2 b}{2 x^{2/3}}-\frac{9 a b^2}{\sqrt [3]{x}}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3/x^2,x]

[Out]

-(a^3/x) - (9*a^2*b)/(2*x^(2/3)) - (9*a*b^2)/x^(1/3) + b^3*Log[x]

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Rubi in Sympy [A]  time = 8.48079, size = 41, normalized size = 1.05 \[ - \frac{a^{3}}{x} - \frac{9 a^{2} b}{2 x^{\frac{2}{3}}} - \frac{9 a b^{2}}{\sqrt [3]{x}} + 3 b^{3} \log{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/x - 9*a**2*b/(2*x**(2/3)) - 9*a*b**2/x**(1/3) + 3*b**3*log(x**(1/3))

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Mathematica [A]  time = 0.0318332, size = 40, normalized size = 1.03 \[ b^3 \log (x)-\frac{a \left (2 a^2+9 a b \sqrt [3]{x}+18 b^2 x^{2/3}\right )}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(2*a^2 + 9*a*b*x^(1/3) + 18*b^2*x^(2/3)))/(2*x) + b^3*Log[x]

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Maple [A]  time = 0.009, size = 34, normalized size = 0.9 \[ -{\frac{{a}^{3}}{x}}-{\frac{9\,{a}^{2}b}{2}{x}^{-{\frac{2}{3}}}}-9\,{\frac{a{b}^{2}}{\sqrt [3]{x}}}+{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3/x^2,x)

[Out]

-a^3/x-9/2*a^2*b/x^(2/3)-9*a*b^2/x^(1/3)+b^3*ln(x)

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Maxima [A]  time = 1.4295, size = 49, normalized size = 1.26 \[ b^{3} \log \left (x\right ) - \frac{18 \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} + 2 \, a^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*log(x) - 1/2*(18*a*b^2*x^(2/3) + 9*a^2*b*x^(1/3) + 2*a^3)/x

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Fricas [A]  time = 0.219189, size = 53, normalized size = 1.36 \[ \frac{6 \, b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 18 \, a b^{2} x^{\frac{2}{3}} - 9 \, a^{2} b x^{\frac{1}{3}} - 2 \, a^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*b^3*x*log(x^(1/3)) - 18*a*b^2*x^(2/3) - 9*a^2*b*x^(1/3) - 2*a^3)/x

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Sympy [A]  time = 2.27901, size = 36, normalized size = 0.92 \[ - \frac{a^{3}}{x} - \frac{9 a^{2} b}{2 x^{\frac{2}{3}}} - \frac{9 a b^{2}}{\sqrt [3]{x}} + b^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/x - 9*a**2*b/(2*x**(2/3)) - 9*a*b**2/x**(1/3) + b**3*log(x)

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GIAC/XCAS [A]  time = 0.21915, size = 50, normalized size = 1.28 \[ b^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{18 \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} + 2 \, a^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*ln(abs(x)) - 1/2*(18*a*b^2*x^(2/3) + 9*a^2*b*x^(1/3) + 2*a^3)/x