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

Optimal. Leaf size=54 \[ -\frac{\log \left (a+b x^3\right )}{3 a^3}+\frac{\log (x)}{a^3}+\frac{1}{3 a^2 \left (a+b x^3\right )}+\frac{1}{6 a \left (a+b x^3\right )^2} \]

[Out]

1/(6*a*(a + b*x^3)^2) + 1/(3*a^2*(a + b*x^3)) + Log[x]/a^3 - Log[a + b*x^3]/(3*a
^3)

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Rubi [A]  time = 0.080797, antiderivative size = 54, 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 \[ -\frac{\log \left (a+b x^3\right )}{3 a^3}+\frac{\log (x)}{a^3}+\frac{1}{3 a^2 \left (a+b x^3\right )}+\frac{1}{6 a \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)^3),x]

[Out]

1/(6*a*(a + b*x^3)^2) + 1/(3*a^2*(a + b*x^3)) + Log[x]/a^3 - Log[a + b*x^3]/(3*a
^3)

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Rubi in Sympy [A]  time = 11.3455, size = 49, normalized size = 0.91 \[ \frac{1}{6 a \left (a + b x^{3}\right )^{2}} + \frac{1}{3 a^{2} \left (a + b x^{3}\right )} + \frac{\log{\left (x^{3} \right )}}{3 a^{3}} - \frac{\log{\left (a + b x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0559465, size = 43, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+2 b x^3\right )}{\left (a+b x^3\right )^2}-2 \log \left (a+b x^3\right )+6 \log (x)}{6 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a*(3*a + 2*b*x^3))/(a + b*x^3)^2 + 6*Log[x] - 2*Log[a + b*x^3])/(6*a^3)

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Maple [A]  time = 0.011, size = 49, normalized size = 0.9 \[{\frac{1}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{1}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.42738, size = 81, normalized size = 1.5 \[ \frac{2 \, b x^{3} + 3 \, a}{6 \,{\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} - \frac{\log \left (b x^{3} + a\right )}{3 \, a^{3}} + \frac{\log \left (x^{3}\right )}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^3 + 3*a)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) - 1/3*log(b*x^3 + a)/a^3 +
1/3*log(x^3)/a^3

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Fricas [A]  time = 0.219627, size = 122, normalized size = 2.26 \[ \frac{2 \, a b x^{3} + 3 \, a^{2} - 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{3} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*a*b*x^3 + 3*a^2 - 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*log(b*x^3 + a) + 6*(b^2*x
^6 + 2*a*b*x^3 + a^2)*log(x))/(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)

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Sympy [A]  time = 3.47081, size = 56, normalized size = 1.04 \[ \frac{3 a + 2 b x^{3}}{6 a^{4} + 12 a^{3} b x^{3} + 6 a^{2} b^{2} x^{6}} + \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**3+a)**3,x)

[Out]

(3*a + 2*b*x**3)/(6*a**4 + 12*a**3*b*x**3 + 6*a**2*b**2*x**6) + log(x)/a**3 - lo
g(a/b + x**3)/(3*a**3)

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GIAC/XCAS [A]  time = 0.228363, size = 77, normalized size = 1.43 \[ -\frac{{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{3 \, b^{2} x^{6} + 8 \, a b x^{3} + 6 \, a^{2}}{6 \,{\left (b x^{3} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(abs(b*x^3 + a))/a^3 + ln(abs(x))/a^3 + 1/6*(3*b^2*x^6 + 8*a*b*x^3 + 6*a^
2)/((b*x^3 + a)^2*a^3)