3.1972 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^4} \, dx\)

Optimal. Leaf size=15 \[ -\frac{\log \left (a+\frac{b}{x^3}\right )}{3 b} \]

[Out]

-Log[a + b/x^3]/(3*b)

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Rubi [A]  time = 0.0181738, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\log \left (a+\frac{b}{x^3}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x^4),x]

[Out]

-Log[a + b/x^3]/(3*b)

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Rubi in Sympy [A]  time = 2.13884, size = 12, normalized size = 0.8 \[ - \frac{\log{\left (a + \frac{b}{x^{3}} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)/x**4,x)

[Out]

-log(a + b/x**3)/(3*b)

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Mathematica [A]  time = 0.00958125, size = 22, normalized size = 1.47 \[ \frac{\log (x)}{b}-\frac{\log \left (a x^3+b\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)*x^4),x]

[Out]

Log[x]/b - Log[b + a*x^3]/(3*b)

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Maple [A]  time = 0.007, size = 21, normalized size = 1.4 \[{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( a{x}^{3}+b \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.42677, size = 18, normalized size = 1.2 \[ -\frac{\log \left (a + \frac{b}{x^{3}}\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^4),x, algorithm="maxima")

[Out]

-1/3*log(a + b/x^3)/b

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Fricas [A]  time = 0.235955, size = 24, normalized size = 1.6 \[ -\frac{\log \left (a x^{3} + b\right ) - 3 \, \log \left (x\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^4),x, algorithm="fricas")

[Out]

-1/3*(log(a*x^3 + b) - 3*log(x))/b

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Sympy [A]  time = 0.60819, size = 15, normalized size = 1. \[ \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (x^{3} + \frac{b}{a} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x**4,x)

[Out]

log(x)/b - log(x**3 + b/a)/(3*b)

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GIAC/XCAS [A]  time = 0.225933, size = 30, normalized size = 2. \[ -\frac{{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^4),x, algorithm="giac")

[Out]

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