3.435 \(\int \frac{a+3 b x^2}{a x+b x^3} \, dx\)

Optimal. Leaf size=10 \[ \log \left (a x+b x^3\right ) \]

[Out]

Log[a*x + b*x^3]

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Rubi [A]  time = 0.0103937, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \log \left (a x+b x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + 3*b*x^2)/(a*x + b*x^3),x]

[Out]

Log[a*x + b*x^3]

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Rubi in Sympy [A]  time = 12.4693, size = 14, normalized size = 1.4 \[ \frac{\log{\left (x^{2} \right )}}{2} + \log{\left (a + b x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*b*x**2+a)/(b*x**3+a*x),x)

[Out]

log(x**2)/2 + log(a + b*x**2)

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Mathematica [A]  time = 0.00922223, size = 11, normalized size = 1.1 \[ \log \left (a+b x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + 3*b*x^2)/(a*x + b*x^3),x]

[Out]

Log[x] + Log[a + b*x^2]

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Maple [A]  time = 0.002, size = 11, normalized size = 1.1 \[ \ln \left ( x \left ( b{x}^{2}+a \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*b*x^2+a)/(b*x^3+a*x),x)

[Out]

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

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Maxima [A]  time = 0.789497, size = 14, normalized size = 1.4 \[ \log \left (b x^{3} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*x^2 + a)/(b*x^3 + a*x),x, algorithm="maxima")

[Out]

log(b*x^3 + a*x)

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Fricas [A]  time = 0.259774, size = 14, normalized size = 1.4 \[ \log \left (b x^{3} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*x^2 + a)/(b*x^3 + a*x),x, algorithm="fricas")

[Out]

log(b*x^3 + a*x)

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Sympy [A]  time = 1.11232, size = 8, normalized size = 0.8 \[ \log{\left (a x + b x^{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*x**2+a)/(b*x**3+a*x),x)

[Out]

log(a*x + b*x**3)

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GIAC/XCAS [A]  time = 0.261366, size = 22, normalized size = 2.2 \[ \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) +{\rm ln}\left ({\left | b x^{2} + a \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*x^2 + a)/(b*x^3 + a*x),x, algorithm="giac")

[Out]

1/2*ln(x^2) + ln(abs(b*x^2 + a))