\(\int \frac {a+b x}{a c-b c x} \, dx\) [1032]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {x}{c}-\frac {2 a \log (a-b x)}{b c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, 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} \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {2 a \log (a-b x)}{b c}-\frac {x}{c} \]

[In]

Int[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

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 (-\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] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {x}{c}-\frac {2 a \log (a-b x)}{b c} \]

[In]

Integrate[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {-x -\frac {2 a \ln \left (-b x +a \right )}{b}}{c}\)	\(22\)
norman	\(-\frac {x}{c}-\frac {2 a \ln \left (-b x +a \right )}{b c}\)	\(24\)
risch	\(-\frac {x}{c}-\frac {2 a \ln \left (-b x +a \right )}{b c}\)	\(24\)
parallelrisch	\(\frac {-2 a \ln \left (b x -a \right )-b x}{b c}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {b x + 2 \, a \log \left (b x - a\right )}{b c} \]

[In]

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

[Out]

-(b*x + 2*a*log(b*x - a))/(b*c)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x}{a c-b c x} \, dx=- \frac {2 a \log {\left (- a + b x \right )}}{b c} - \frac {x}{c} \]

[In]

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

[Out]

-2*a*log(-a + b*x)/(b*c) - x/c

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {x}{c} - \frac {2 \, a \log \left (b x - a\right )}{b c} \]

[In]

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

[Out]

-x/c - 2*a*log(b*x - a)/(b*c)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {x}{c} - \frac {2 \, a \log \left ({\left | b x - a \right |}\right )}{b c} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{a c-b c x} \, dx=-\frac {b\,x+2\,a\,\ln \left (b\,x-a\right )}{b\,c} \]

[In]

int((a + b*x)/(a*c - b*c*x),x)

[Out]

-(b*x + 2*a*log(b*x - a))/(b*c)