Optimal. Leaf size=43 \[ -\frac{\log (x) (A b-a B)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]
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Rubi [A] time = 0.0278808, antiderivative size = 43, 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 = {77} \[ -\frac{\log (x) (A b-a B)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^2 (a+b x)} \, dx &=\int \left (\frac{A}{a x^2}+\frac{-A b+a B}{a^2 x}-\frac{b (-A b+a B)}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac{A}{a x}-\frac{(A b-a B) \log (x)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0167456, size = 42, normalized size = 0.98 \[ \frac{\log (x) (a B-A b)}{a^2}+\frac{(A b-a B) \log (a+b x)}{a^2}-\frac{A}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 51, normalized size = 1.2 \begin{align*} -{\frac{A}{ax}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{a}}+{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ) B}{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99499, size = 58, normalized size = 1.35 \begin{align*} -\frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{a^{2}} + \frac{{\left (B a - A b\right )} \log \left (x\right )}{a^{2}} - \frac{A}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81509, size = 92, normalized size = 2.14 \begin{align*} -\frac{{\left (B a - A b\right )} x \log \left (b x + a\right ) -{\left (B a - A b\right )} x \log \left (x\right ) + A a}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.625741, size = 95, normalized size = 2.21 \begin{align*} - \frac{A}{a x} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b + B a^{2} - a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} - \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b + B a^{2} + a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21402, size = 69, normalized size = 1.6 \begin{align*} \frac{{\left (B a - A b\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{A}{a x} - \frac{{\left (B a b - A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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