Optimal. Leaf size=62 \[ \frac{b B-A c}{b^2 (b+c x)}+\frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}-\frac{A}{b^2 x} \]
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Rubi [A] time = 0.0501146, antiderivative size = 62, 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 = {631} \[ \frac{b B-A c}{b^2 (b+c x)}+\frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}-\frac{A}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{A}{b^2 x^2}+\frac{b B-2 A c}{b^3 x}-\frac{c (b B-A c)}{b^2 (b+c x)^2}-\frac{c (b B-2 A c)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{A}{b^2 x}+\frac{b B-A c}{b^2 (b+c x)}+\frac{(b B-2 A c) \log (x)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0465392, size = 56, normalized size = 0.9 \[ \frac{\frac{b (b B-A c)}{b+c x}+\log (x) (b B-2 A c)+(2 A c-b B) \log (b+c x)-\frac{A b}{x}}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 78, normalized size = 1.3 \begin{align*} -{\frac{A}{{b}^{2}x}}-2\,{\frac{Ac\ln \left ( x \right ) }{{b}^{3}}}+{\frac{\ln \left ( x \right ) B}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Ac}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) B}{{b}^{2}}}-{\frac{Ac}{{b}^{2} \left ( cx+b \right ) }}+{\frac{B}{b \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12109, size = 90, normalized size = 1.45 \begin{align*} -\frac{A b -{\left (B b - 2 \, A c\right )} x}{b^{2} c x^{2} + b^{3} x} - \frac{{\left (B b - 2 \, A c\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (B b - 2 \, A c\right )} \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16945, size = 227, normalized size = 3.66 \begin{align*} -\frac{A b^{2} -{\left (B b^{2} - 2 \, A b c\right )} x +{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (c x + b\right ) -{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.697171, size = 128, normalized size = 2.06 \begin{align*} \frac{- A b + x \left (- 2 A c + B b\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} - b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} - \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} + b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24799, size = 96, normalized size = 1.55 \begin{align*} \frac{{\left (B b - 2 \, A c\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{B b x - 2 \, A c x - A b}{{\left (c x^{2} + b x\right )} b^{2}} - \frac{{\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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