3.1058 \(\int \frac{a c-b c x}{(a+b x)^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]

[Out]

(-2*a*c)/(b*(a + b*x)) - (c*Log[a + b*x])/b

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Rubi [A]  time = 0.0338715, antiderivative size = 27, 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 \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*c)/(b*(a + b*x)) - (c*Log[a + b*x])/b

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Rubi in Sympy [A]  time = 8.59556, size = 22, normalized size = 0.81 \[ - \frac{2 a c}{b \left (a + b x\right )} - \frac{c \log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0143324, size = 23, normalized size = 0.85 \[ -\frac{c \left (\frac{2 a}{a+b x}+\log (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

-((c*((2*a)/(a + b*x) + Log[a + b*x]))/b)

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Maple [A]  time = 0.01, size = 28, normalized size = 1. \[ -2\,{\frac{ac}{b \left ( bx+a \right ) }}-{\frac{c\ln \left ( bx+a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*c*x+a*c)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35452, size = 38, normalized size = 1.41 \[ -\frac{2 \, a c}{b^{2} x + a b} - \frac{c \log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199769, size = 45, normalized size = 1.67 \[ -\frac{2 \, a c +{\left (b c x + a c\right )} \log \left (b x + a\right )}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.28021, size = 24, normalized size = 0.89 \[ - \frac{2 a c}{a b + b^{2} x} - \frac{c \log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206808, size = 73, normalized size = 2.7 \[ c{\left (\frac{{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )} - \frac{a c}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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