3.2353 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a} \]

[Out]

(-3*Log[a + b*x^(1/3)])/a + Log[x]/a

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Rubi [A]  time = 0.0315545, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{a}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))*x),x]

[Out]

(-3*Log[a + b*x^(1/3)])/a + Log[x]/a

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Rubi in Sympy [A]  time = 6.58708, size = 22, normalized size = 1. \[ \frac{3 \log{\left (\sqrt [3]{x} \right )}}{a} - \frac{3 \log{\left (a + b \sqrt [3]{x} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00800117, size = 27, normalized size = 1.23 \[ \frac{3 \log \left (\sqrt [3]{x}\right )}{a}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 21, normalized size = 1. \[ -3\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{a}}+{\frac{\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))/x,x)

[Out]

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

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Maxima [A]  time = 1.44184, size = 27, normalized size = 1.23 \[ -\frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(b*x^(1/3) + a)/a + log(x)/a

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Fricas [A]  time = 0.22336, size = 27, normalized size = 1.23 \[ -\frac{3 \,{\left (\log \left (b x^{\frac{1}{3}} + a\right ) - \log \left (x^{\frac{1}{3}}\right )\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.60658, size = 37, normalized size = 1.68 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt [3]{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{3}{b \sqrt [3]{x}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{3 \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**(1/3))/x,x)

[Out]

Piecewise((zoo/x**(1/3), Eq(a, 0) & Eq(b, 0)), (log(x)/a, Eq(b, 0)), (-3/(b*x**(
1/3)), Eq(a, 0)), (log(x)/a - 3*log(a/b + x**(1/3))/a, True))

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GIAC/XCAS [A]  time = 0.21826, size = 30, normalized size = 1.36 \[ -\frac{3 \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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