3.164 \(\int \frac{b+2 c x}{b x+c x^2} \, dx\)

Optimal. Leaf size=10 \[ \log \left (b x+c x^2\right ) \]

[Out]

Log[b*x + c*x^2]

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Rubi [A]  time = 0.00952077, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \log \left (b x+c x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[b*x + c*x^2]

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Rubi in Sympy [A]  time = 3.22386, size = 8, normalized size = 0.8 \[ \log{\left (b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x + c*x**2)

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Mathematica [A]  time = 0.00619487, size = 9, normalized size = 0.9 \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

Log[x] + Log[b + c*x]

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Maple [A]  time = 0.001, size = 9, normalized size = 0.9 \[ \ln \left ( x \left ( cx+b \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x),x)

[Out]

ln(x*(c*x+b))

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Maxima [A]  time = 0.804943, size = 14, normalized size = 1.4 \[ \log \left (c x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^2 + b*x)

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Fricas [A]  time = 0.250039, size = 14, normalized size = 1.4 \[ \log \left (c x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^2 + b*x)

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Sympy [A]  time = 1.0744, size = 8, normalized size = 0.8 \[ \log{\left (b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.260005, size = 15, normalized size = 1.5 \[{\rm ln}\left ({\left | c x + b \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(c*x + b)) + ln(abs(x))