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

Optimal. Leaf size=25 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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Rubi [A]  time = 0.01925, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x),x]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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

Mathematica [A]  time = 0.006734, size = 25, normalized size = 1. \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x),x]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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Maple [A]  time = 0., size = 32, normalized size = 1.3 \begin{align*}{\frac{dx}{b}}-{\frac{a\ln \left ( bx+a \right ) d}{{b}^{2}}}+{\frac{c\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0742, size = 34, normalized size = 1.36 \begin{align*} \frac{d x}{b} + \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x/b + (b*c - a*d)*log(b*x + a)/b^2

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Fricas [A]  time = 1.87439, size = 54, normalized size = 2.16 \begin{align*} \frac{b d x +{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.486488, size = 20, normalized size = 0.8 \begin{align*} \frac{d x}{b} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x/b - (a*d - b*c)*log(a + b*x)/b**2

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Giac [A]  time = 1.17339, size = 35, normalized size = 1.4 \begin{align*} \frac{d x}{b} + \frac{{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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