3.1051 \(\int \frac{a c-b c x}{a+b x} \, dx\)

Optimal. Leaf size=18 \[ \frac{2 a c \log (a+b x)}{b}-c x \]

[Out]

-(c*x) + (2*a*c*Log[a + b*x])/b

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Rubi [A]  time = 0.0103672, antiderivative size = 18, 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} \[ \frac{2 a c \log (a+b x)}{b}-c x \]

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)/(a + b*x),x]

[Out]

-(c*x) + (2*a*c*Log[a + b*x])/b

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

Mathematica [A]  time = 0.0039915, size = 18, normalized size = 1. \[ c \left (\frac{2 a \log (a+b x)}{b}-x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c - b*c*x)/(a + b*x),x]

[Out]

c*(-x + (2*a*Log[a + b*x])/b)

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Maple [A]  time = 0.001, size = 19, normalized size = 1.1 \begin{align*} -cx+2\,{\frac{ac\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*c*x+a*c)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.0294, size = 24, normalized size = 1.33 \begin{align*} -c x + \frac{2 \, a c \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*x + 2*a*c*log(b*x + a)/b

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Fricas [A]  time = 1.41083, size = 45, normalized size = 2.5 \begin{align*} -\frac{b c x - 2 \, a c \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c*x - 2*a*c*log(b*x + a))/b

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Sympy [A]  time = 0.276684, size = 15, normalized size = 0.83 \begin{align*} \frac{2 a c \log{\left (a + b x \right )}}{b} - c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*c*log(a + b*x)/b - c*x

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Giac [A]  time = 1.04863, size = 26, normalized size = 1.44 \begin{align*} -c x + \frac{2 \, a c \log \left ({\left | b x + a \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*x + 2*a*c*log(abs(b*x + a))/b