3.338 \(\int \frac{1}{\frac{b}{x}+a x} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

[Out]

Log[b + a*x^2]/(2*a)

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Rubi [A]  time = 0.00984524, 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 \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

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Rubi in Sympy [A]  time = 2.48597, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{2} + b \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b/x+a*x),x)

[Out]

log(a*x**2 + b)/(2*a)

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Mathematica [A]  time = 0.00362829, size = 15, normalized size = 1. \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x+a*x),x)

[Out]

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

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Maxima [A]  time = 1.3831, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{2} + b\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(a*x^2 + b)/a

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Fricas [A]  time = 0.214143, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{2} + b\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(a*x^2 + b)/a

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Sympy [A]  time = 0.236194, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{2} + b \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x+a*x),x)

[Out]

log(a*x**2 + b)/(2*a)

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GIAC/XCAS [A]  time = 0.216392, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(a*x^2 + b))/a