3.2425 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x} \, dx\)

Optimal. Leaf size=33 \[ \frac{3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0604464, antiderivative size = 33, 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 \[ \frac{3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))^2*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.14488, size = 29, normalized size = 0.88 \[ \frac{3 b}{a^{2} \left (a \sqrt [3]{x} + b\right )} + \frac{3 \log{\left (a \sqrt [3]{x} + b \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0213829, size = 29, normalized size = 0.88 \[ \frac{3 \left (\frac{b}{a \sqrt [3]{x}+b}+\log \left (a \sqrt [3]{x}+b\right )\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 0.9 \[ 3\,{\frac{b}{{a}^{2} \left ( b+a\sqrt [3]{x} \right ) }}+3\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2/x,x)

[Out]

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

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Maxima [A]  time = 1.43999, size = 46, normalized size = 1.39 \[ -\frac{3}{a^{2} + \frac{a b}{x^{\frac{1}{3}}}} + \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229404, size = 47, normalized size = 1.42 \[ \frac{3 \,{\left ({\left (a x^{\frac{1}{3}} + b\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + b\right )}}{a^{3} x^{\frac{1}{3}} + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.9554, size = 99, normalized size = 3. \[ \begin{cases} \frac{3 a x \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} + \frac{3 b x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} + \frac{3 b x^{\frac{2}{3}}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{2}{3}}}{2 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((3*a*x*log(x**(1/3) + b/a)/(a**3*x + a**2*b*x**(2/3)) + 3*b*x**(2/3)*l
og(x**(1/3) + b/a)/(a**3*x + a**2*b*x**(2/3)) + 3*b*x**(2/3)/(a**3*x + a**2*b*x*
*(2/3)), Ne(a, 0)), (3*x**(2/3)/(2*b**2), True))

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GIAC/XCAS [A]  time = 0.214898, size = 41, normalized size = 1.24 \[ \frac{3 \,{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{2}} + \frac{3 \, b}{{\left (a x^{\frac{1}{3}} + b\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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