\(\int \frac {c+d x}{x (a+b x)} \, dx\) [209]

Optimal result

Integrand size = 16, antiderivative size = 30 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b} \]

[In]

Int[(c + d*x)/(x*(a + b*x)),x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{a x}+\frac {-b c+a d}{a (a+b x)}\right ) \, dx \\ & = \frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log (x)}{a}+\frac {(-b c+a d) \log (a+b x)}{a b} \]

[In]

Integrate[(c + d*x)/(x*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {c \ln \left (x \right )}{a}+\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{a b}\)	\(30\)
norman	\(\frac {c \ln \left (x \right )}{a}+\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{a b}\)	\(30\)
parallelrisch	\(\frac {c \ln \left (x \right ) b +\ln \left (b x +a \right ) a d -\ln \left (b x +a \right ) b c}{a b}\)	\(33\)
risch	\(\frac {c \ln \left (x \right )}{a}+\frac {\ln \left (-b x -a \right ) d}{b}-\frac {\ln \left (-b x -a \right ) c}{a}\)	\(38\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {b c \log \left (x\right ) - {\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log {\left (x \right )}}{a} + \frac {\left (a d - b c\right ) \log {\left (x + \frac {- a c + \frac {a \left (a d - b c\right )}{b}}{a d - 2 b c} \right )}}{a b} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log \left (x\right )}{a} - \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c \log \left ({\left | x \right |}\right )}{a} - \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \]

[In]

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

[Out]

c*log(abs(x))/a - (b*c - a*d)*log(abs(b*x + a))/(a*b)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x}{x (a+b x)} \, dx=\frac {c\,\ln \left (x\right )}{a}-\ln \left (a+b\,x\right )\,\left (\frac {c}{a}-\frac {d}{b}\right ) \]

[In]

int((c + d*x)/(x*(a + b*x)),x)

[Out]

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