Optimal. Leaf size=27 \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0132458, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{a c-b c x}{(a+b x)^2} \, dx &=\int \left (\frac{2 a c}{(a+b x)^2}-\frac{c}{a+b x}\right ) \, dx\\ &=-\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.009045, size = 23, normalized size = 0.85 \[ -\frac{c \left (\frac{2 a}{a+b x}+\log (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 28, normalized size = 1. \begin{align*} -2\,{\frac{ac}{b \left ( bx+a \right ) }}-{\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 = 1.02968, size = 38, normalized size = 1.41 \begin{align*} -\frac{2 \, a c}{b^{2} x + a b} - \frac{c \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53493, size = 72, normalized size = 2.67 \begin{align*} -\frac{2 \, a c +{\left (b c x + a c\right )} \log \left (b x + a\right )}{b^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.318086, size = 24, normalized size = 0.89 \begin{align*} - \frac{2 a c}{a b + b^{2} x} - \frac{c \log{\left (a + b x \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.05256, size = 73, normalized size = 2.7 \begin{align*} c{\left (\frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )} - \frac{a c}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]