Optimal. Leaf size=25 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 43
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]