Optimal. Leaf size=32 \[ \frac{B \log (a+b x)}{b^2}-\frac{A b-a B}{b^2 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0217468, antiderivative size = 32, 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} \[ \frac{B \log (a+b x)}{b^2}-\frac{A b-a B}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^2} \, dx &=\int \left (\frac{A b-a B}{b (a+b x)^2}+\frac{B}{b (a+b x)}\right ) \, dx\\ &=-\frac{A b-a B}{b^2 (a+b x)}+\frac{B \log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.010522, size = 31, normalized size = 0.97 \[ \frac{a B-A b}{b^2 (a+b x)}+\frac{B \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 39, normalized size = 1.2 \begin{align*}{\frac{B\ln \left ( bx+a \right ) }{{b}^{2}}}-{\frac{A}{b \left ( bx+a \right ) }}+{\frac{Ba}{{b}^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03896, size = 46, normalized size = 1.44 \begin{align*} \frac{B a - A b}{b^{3} x + a b^{2}} + \frac{B \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 = 1.57666, size = 78, normalized size = 2.44 \begin{align*} \frac{B a - A b +{\left (B b x + B a\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.338427, size = 27, normalized size = 0.84 \begin{align*} \frac{B \log{\left (a + b x \right )}}{b^{2}} + \frac{- A b + B a}{a b^{2} + b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.96977, size = 77, normalized size = 2.41 \begin{align*} -\frac{B{\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 )}}{b} - \frac{A}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]