3.2362 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2 x} \, dx\)

Optimal. Leaf size=38 \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{3}{a \left (a+b \sqrt [3]{x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0571982, antiderivative size = 38, 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 \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{3}{a \left (a+b \sqrt [3]{x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))^2*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.6017, size = 37, normalized size = 0.97 \[ \frac{3}{a \left (a + b \sqrt [3]{x}\right )} + \frac{3 \log{\left (\sqrt [3]{x} \right )}}{a^{2}} - \frac{3 \log{\left (a + b \sqrt [3]{x} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0499961, size = 37, normalized size = 0.97 \[ \frac{3 \left (\frac{a}{a+b \sqrt [3]{x}}-\log \left (a+b \sqrt [3]{x}\right )+\frac{\log (x)}{3}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \[ 3\,{\frac{1}{a \left ( a+b\sqrt [3]{x} \right ) }}-3\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{2}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^2/x,x)

[Out]

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

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Maxima [A]  time = 1.44099, size = 46, normalized size = 1.21 \[ \frac{3}{a b x^{\frac{1}{3}} + a^{2}} - \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225277, size = 66, normalized size = 1.74 \[ -\frac{3 \,{\left ({\left (b x^{\frac{1}{3}} + a\right )} \log \left (b x^{\frac{1}{3}} + a\right ) -{\left (b x^{\frac{1}{3}} + a\right )} \log \left (x^{\frac{1}{3}}\right ) - a\right )}}{a^{2} b x^{\frac{1}{3}} + a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.596, size = 160, normalized size = 4.21 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{3}{2 b^{2} x^{\frac{2}{3}}} & \text{for}\: a = 0 \\\frac{a x^{\frac{2}{3}} \log{\left (x \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} - \frac{3 a x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} + \frac{3 a x^{\frac{2}{3}}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} + \frac{b x \log{\left (x \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} - \frac{3 b x \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(2/3), Eq(a, 0) & Eq(b, 0)), (log(x)/a**2, Eq(b, 0)), (-3/(2*b
**2*x**(2/3)), Eq(a, 0)), (a*x**(2/3)*log(x)/(a**3*x**(2/3) + a**2*b*x) - 3*a*x*
*(2/3)*log(a/b + x**(1/3))/(a**3*x**(2/3) + a**2*b*x) + 3*a*x**(2/3)/(a**3*x**(2
/3) + a**2*b*x) + b*x*log(x)/(a**3*x**(2/3) + a**2*b*x) - 3*b*x*log(a/b + x**(1/
3))/(a**3*x**(2/3) + a**2*b*x), True))

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GIAC/XCAS [A]  time = 0.220957, size = 49, normalized size = 1.29 \[ -\frac{3 \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{3}{{\left (b x^{\frac{1}{3}} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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