3.209 \(\int \frac{c+d x}{x (a+b x)} \, dx\)

Optimal. Leaf size=30 \[ \frac{c \log (x)}{a}-\frac{(b c-a d) \log (a+b x)}{a b} \]

[Out]

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

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Rubi [A]  time = 0.0182764, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {72} \[ \frac{c \log (x)}{a}-\frac{(b c-a d) \log (a+b x)}{a b} \]

Antiderivative was successfully verified.

[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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{c+d x}{x (a+b x)} \, dx &=\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]  time = 0.0101575, size = 29, normalized size = 0.97 \[ \frac{(a d-b c) \log (a+b x)}{a b}+\frac{c \log (x)}{a} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.004, size = 32, normalized size = 1.1 \begin{align*}{\frac{c\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( bx+a \right ) d}{b}}-{\frac{\ln \left ( bx+a \right ) c}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05905, size = 41, normalized size = 1.37 \begin{align*} \frac{c \log \left (x\right )}{a} - \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.95524, size = 63, normalized size = 2.1 \begin{align*} \frac{b c \log \left (x\right ) -{\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.496797, size = 41, normalized size = 1.37 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.20145, size = 43, normalized size = 1.43 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)