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 = 0.012133, 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, Rules used = {43} \[ -\frac{2 a \log (a-b x)}{b c}-\frac{x}{c} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{a c-b c x} \, dx &=\int \left (-\frac{1}{c}+\frac{2 a}{c (a-b x)}\right ) \, dx\\ &=-\frac{x}{c}-\frac{2 a \log (a-b x)}{b c}\\ \end{align*}

Mathematica [A]  time = 0.0048164, 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 = 0.002, size = 25, normalized size = 1.1 \begin{align*} -{\frac{x}{c}}-2\,{\frac{a\ln \left ( bx-a \right ) }{bc}} \end{align*}

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

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Maxima [A]  time = 1.02062, size = 32, normalized size = 1.39 \begin{align*} -\frac{x}{c} - \frac{2 \, a \log \left (b x - a\right )}{b c} \end{align*}

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 = 1.47799, size = 45, normalized size = 1.96 \begin{align*} -\frac{b x + 2 \, a \log \left (b x - a\right )}{b c} \end{align*}

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 = 0.285939, size = 17, normalized size = 0.74 \begin{align*} - \frac{2 a \log{\left (- a + b x \right )}}{b c} - \frac{x}{c} \end{align*}

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 [A]  time = 1.05307, size = 34, normalized size = 1.48 \begin{align*} -\frac{x}{c} - \frac{2 \, a \log \left ({\left | b x - a \right |}\right )}{b c} \end{align*}

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*log(abs(b*x - a))/(b*c)