3.1032 \(\int \frac{a+b x}{a c-b c x} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 a \log (a-b x)}{b c}-\frac{x}{c} \]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

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Rubi [A]  time = 0.0316786, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 a \log (a-b x)}{b c}-\frac{x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \log{\left (a - b x \right )}}{b c} - \int \frac{1}{c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*log(a - b*x)/(b*c) - Integral(1/c, x)

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Mathematica [A]  time = 0.00663997, size = 23, normalized size = 1. \[ -\frac{2 a \log (a-b x)}{b c}-\frac{x}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.1 \[ -{\frac{x}{c}}-2\,{\frac{a\ln \left ( bx-a \right ) }{bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b*c*x+a*c),x)

[Out]

-x/c-2/c*a/b*ln(b*x-a)

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Maxima [A]  time = 1.33441, size = 32, normalized size = 1.39 \[ -\frac{x}{c} - \frac{2 \, a \log \left (b x - a\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/c - 2*a*log(b*x - a)/(b*c)

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Fricas [A]  time = 0.198649, size = 31, normalized size = 1.35 \[ -\frac{b x + 2 \, a \log \left (b x - a\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x + 2*a*log(b*x - a))/(b*c)

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Sympy [A]  time = 1.11306, size = 17, normalized size = 0.74 \[ - \frac{2 a \log{\left (- a + b x \right )}}{b c} - \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*log(-a + b*x)/(b*c) - x/c

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GIAC/XCAS [A]  time = 0.203925, size = 34, normalized size = 1.48 \[ -\frac{x}{c} - \frac{2 \, a{\rm ln}\left ({\left | b x - a \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/c - 2*a*ln(abs(b*x - a))/(b*c)