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

Optimal. Leaf size=15 \[ \frac{\log \left (a x^4+b\right )}{4 a} \]

[Out]

Log[b + a*x^4]/(4*a)

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Rubi [A]  time = 0.00945454, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\log \left (a x^4+b\right )}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[(b/x^3 + a*x)^(-1),x]

[Out]

Log[b + a*x^4]/(4*a)

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Rubi in Sympy [A]  time = 2.54026, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{4} + b \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x**4 + b)/(4*a)

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Mathematica [A]  time = 0.00686364, size = 15, normalized size = 1. \[ \frac{\log \left (a x^4+b\right )}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x^3 + a*x)^(-1),x]

[Out]

Log[b + a*x^4]/(4*a)

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( a{x}^{4}+b \right ) }{4\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x^3+a*x),x)

[Out]

1/4*ln(a*x^4+b)/a

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Maxima [A]  time = 1.37943, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{4} + b\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(a*x^4 + b)/a

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Fricas [A]  time = 0.209262, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{4} + b\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(a*x^4 + b)/a

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Sympy [A]  time = 0.365069, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{4} + b \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x**3+a*x),x)

[Out]

log(a*x**4 + b)/(4*a)

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GIAC/XCAS [A]  time = 0.218496, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | a x^{4} + b \right |}\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(abs(a*x^4 + b))/a