3.332 \(\int \frac{1}{x \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^3\right )}{3 a^2}+\frac{\log (x)}{a^2}+\frac{1}{3 a \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0612067, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+b x^3\right )}{3 a^2}+\frac{\log (x)}{a^2}+\frac{1}{3 a \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.23427, size = 34, normalized size = 0.89 \[ \frac{1}{3 a \left (a + b x^{3}\right )} + \frac{\log{\left (x^{3} \right )}}{3 a^{2}} - \frac{\log{\left (a + b x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0226193, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^3}-\log \left (a+b x^3\right )+3 \log (x)}{3 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 35, normalized size = 0.9 \[{\frac{1}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.44106, size = 50, normalized size = 1.32 \[ \frac{1}{3 \,{\left (a b x^{3} + a^{2}\right )}} - \frac{\log \left (b x^{3} + a\right )}{3 \, a^{2}} + \frac{\log \left (x^{3}\right )}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22826, size = 63, normalized size = 1.66 \[ -\frac{{\left (b x^{3} + a\right )} \log \left (b x^{3} + a\right ) - 3 \,{\left (b x^{3} + a\right )} \log \left (x\right ) - a}{3 \,{\left (a^{2} b x^{3} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.9075, size = 34, normalized size = 0.89 \[ \frac{1}{3 a^{2} + 3 a b x^{3}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**3+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224295, size = 61, normalized size = 1.61 \[ -\frac{{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} + 2 \, a}{3 \,{\left (b x^{3} + a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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