\(\int \frac {a c-b c x}{a+b x} \, dx\) [1051]

Optimal result

Integrand size = 17, antiderivative size = 18 \[ \int \frac {a c-b c x}{a+b x} \, dx=-c x+\frac {2 a c \log (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {a c-b c x}{a+b x} \, dx=\frac {2 a c \log (a+b x)}{b}-c x \]

[In]

Int[(a*c - b*c*x)/(a + b*x),x]

[Out]

-(c*x) + (2*a*c*Log[a + b*x])/b

Rule 45

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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a c-b c x}{a+b x} \, dx=c \left (-x+\frac {2 a \log (a+b x)}{b}\right ) \]

[In]

Integrate[(a*c - b*c*x)/(a + b*x),x]

[Out]

c*(-x + (2*a*Log[a + b*x])/b)

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
default	\(c \left (-x +\frac {2 a \ln \left (b x +a \right )}{b}\right )\)	\(19\)
norman	\(-c x +\frac {2 a c \ln \left (b x +a \right )}{b}\)	\(19\)
risch	\(-c x +\frac {2 a c \ln \left (b x +a \right )}{b}\)	\(19\)
parallelrisch	\(\frac {2 a c \ln \left (b x +a \right )-b c x}{b}\)	\(21\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a c-b c x}{a+b x} \, dx=-\frac {b c x - 2 \, a c \log \left (b x + a\right )}{b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {a c-b c x}{a+b x} \, dx=\frac {2 a c \log {\left (a + b x \right )}}{b} - c x \]

[In]

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

[Out]

2*a*c*log(a + b*x)/b - c*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a c-b c x}{a+b x} \, dx=-c x + \frac {2 \, a c \log \left (b x + a\right )}{b} \]

[In]

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

[Out]

-c*x + 2*a*c*log(b*x + a)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {a c-b c x}{a+b x} \, dx=-c x + \frac {2 \, a c \log \left ({\left | b x + a \right |}\right )}{b} \]

[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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a c-b c x}{a+b x} \, dx=\frac {2\,a\,c\,\ln \left (a+b\,x\right )}{b}-c\,x \]

[In]

int((a*c - b*c*x)/(a + b*x),x)

[Out]

(2*a*c*log(a + b*x))/b - c*x