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

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

[Out]

Log[a*x + b*x^2]

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Rubi [A]  time = 0.0104052, 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 \[ \log \left (a x+b x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[a*x + b*x^2]

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Rubi in Sympy [A]  time = 3.2558, size = 8, normalized size = 0.8 \[ \log{\left (a x + b x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b*x**2)

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Mathematica [A]  time = 0.0063123, size = 9, normalized size = 0.9 \[ \log (a+b x)+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

Log[x] + Log[a + b*x]

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Maple [A]  time = 0., size = 9, normalized size = 0.9 \[ \ln \left ( x \left ( bx+a \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*b*x+a)/(b*x^2+a*x),x)

[Out]

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

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Maxima [A]  time = 0.730641, size = 14, normalized size = 1.4 \[ \log \left (b x^{2} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^2 + a*x)

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Fricas [A]  time = 0.214917, size = 14, normalized size = 1.4 \[ \log \left (b x^{2} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^2 + a*x)

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Sympy [A]  time = 1.07752, size = 8, normalized size = 0.8 \[ \log{\left (a x + b x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b*x**2)

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GIAC/XCAS [A]  time = 0.206586, size = 15, normalized size = 1.5 \[{\rm ln}\left ({\left | b x + a \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x + a)) + ln(abs(x))