Optimal. Leaf size=30 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0184414, 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} \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B x}{x (a+b x)} \, dx &=\int \left (\frac{A}{a x}+\frac{-A b+a B}{a (a+b x)}\right ) \, dx\\ &=\frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b}\\ \end{align*}
Mathematica [A] time = 0.0097053, size = 29, normalized size = 0.97 \[ \frac{(a B-A b) \log (a+b x)}{a b}+\frac{A \log (x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 32, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( bx+a \right ) A}{a}}+{\frac{\ln \left ( bx+a \right ) B}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01215, size = 39, normalized size = 1.3 \begin{align*} \frac{A \log \left (x\right )}{a} + \frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62748, size = 63, normalized size = 2.1 \begin{align*} \frac{A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.528548, size = 41, normalized size = 1.37 \begin{align*} \frac{A \log{\left (x \right )}}{a} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a + \frac{a \left (- A b + B a\right )}{b}}{- 2 A b + B a} \right )}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24199, size = 42, normalized size = 1.4 \begin{align*} \frac{A \log \left ({\left | x \right |}\right )}{a} + \frac{{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]