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

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]

[Out]

Log[x]/a - Log[a + b*x^2]/(2*a)

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Rubi [A]  time = 0.03242, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^(-1),x]

[Out]

Log[x]/a - Log[a + b*x^2]/(2*a)

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Rubi in Sympy [A]  time = 158.157, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{2} \right )}}{2 a} - \frac{\log{\left (a + b x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0066918, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^3)^(-1),x]

[Out]

Log[x]/a - Log[a + b*x^2]/(2*a)

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Maple [A]  time = 0.005, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a*x),x)

[Out]

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

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Maxima [A]  time = 1.37395, size = 27, normalized size = 1.23 \[ -\frac{\log \left (b x^{2} + a\right )}{2 \, a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(b*x^2 + a)/a + log(x)/a

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Fricas [A]  time = 0.206427, size = 24, normalized size = 1.09 \[ -\frac{\log \left (b x^{2} + a\right ) - 2 \, \log \left (x\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(log(b*x^2 + a) - 2*log(x))/a

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Sympy [A]  time = 0.498641, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+a*x),x)

[Out]

log(x)/a - log(a/b + x**2)/(2*a)

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GIAC/XCAS [A]  time = 0.218236, size = 32, normalized size = 1.45 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a} - \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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