3.2438 \(\int \frac{1}{1+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=44 \[ -3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)+3 b^2 \sqrt [3]{x}-\frac{3}{2} b x^{2/3}+x \]

[Out]

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

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Rubi [A]  time = 0.0595172, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -3 b^3 \log \left (\frac{b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)+3 b^2 \sqrt [3]{x}-\frac{3}{2} b x^{2/3}+x \]

Antiderivative was successfully verified.

[In]  Int[(1 + b/x^(1/3))^(-1),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b**3*log(b + x**(1/3)) - 3*b*Integral(x, (x, x**(1/3))) + x + 3*Integral(b**2
, (x, x**(1/3)))

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Mathematica [A]  time = 0.0127456, size = 35, normalized size = 0.8 \[ -3 b^3 \log \left (b+\sqrt [3]{x}\right )+3 b^2 \sqrt [3]{x}-\frac{3}{2} b x^{2/3}+x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.6 \[ x-{\frac{3\,b}{2}{x}^{{\frac{2}{3}}}}+3\,{b}^{2}\sqrt [3]{x}-3\,{b}^{3}\ln \left ( \sqrt [3]{x}+b \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+b/x^(1/3)),x)

[Out]

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

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Maxima [A]  time = 1.48262, size = 54, normalized size = 1.23 \[ -b^{3} \log \left (x\right ) - 3 \, b^{3} \log \left (\frac{b}{x^{\frac{1}{3}}} + 1\right ) + \frac{1}{2} \,{\left (\frac{6 \, b^{2}}{x^{\frac{2}{3}}} - \frac{3 \, b}{x^{\frac{1}{3}}} + 2\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22868, size = 36, normalized size = 0.82 \[ -3 \, b^{3} \log \left (b + x^{\frac{1}{3}}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.407957, size = 34, normalized size = 0.77 \[ - 3 b^{3} \log{\left (b + \sqrt [3]{x} \right )} + 3 b^{2} \sqrt [3]{x} - \frac{3 b x^{\frac{2}{3}}}{2} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+b/x**(1/3)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213372, size = 38, normalized size = 0.86 \[ -3 \, b^{3}{\rm ln}\left ({\left | b + x^{\frac{1}{3}} \right |}\right ) + 3 \, b^{2} x^{\frac{1}{3}} - \frac{3}{2} \, b x^{\frac{2}{3}} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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