Optimal. Leaf size=109 \[ -\frac{b B-3 A c}{b^4 x}-\frac{c (2 b B-3 A c)}{b^4 (b+c x)}-\frac{c (b B-A c)}{2 b^3 (b+c x)^2}-\frac{3 c \log (x) (b B-2 A c)}{b^5}+\frac{3 c (b B-2 A c) \log (b+c x)}{b^5}-\frac{A}{2 b^3 x^2} \]
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Rubi [A] time = 0.0986541, antiderivative size = 109, 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} \[ -\frac{b B-3 A c}{b^4 x}-\frac{c (2 b B-3 A c)}{b^4 (b+c x)}-\frac{c (b B-A c)}{2 b^3 (b+c x)^2}-\frac{3 c \log (x) (b B-2 A c)}{b^5}+\frac{3 c (b B-2 A c) \log (b+c x)}{b^5}-\frac{A}{2 b^3 x^2} \]
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 )^3} \, dx &=\int \left (\frac{A}{b^3 x^3}+\frac{b B-3 A c}{b^4 x^2}-\frac{3 c (b B-2 A c)}{b^5 x}+\frac{c^2 (b B-A c)}{b^3 (b+c x)^3}+\frac{c^2 (2 b B-3 A c)}{b^4 (b+c x)^2}+\frac{3 c^2 (b B-2 A c)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{A}{2 b^3 x^2}-\frac{b B-3 A c}{b^4 x}-\frac{c (b B-A c)}{2 b^3 (b+c x)^2}-\frac{c (2 b B-3 A c)}{b^4 (b+c x)}-\frac{3 c (b B-2 A c) \log (x)}{b^5}+\frac{3 c (b B-2 A c) \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0819153, size = 106, normalized size = 0.97 \[ \frac{-\frac{b \left (A \left (-4 b^2 c x+b^3-18 b c^2 x^2-12 c^3 x^3\right )+b B x \left (2 b^2+9 b c x+6 c^2 x^2\right )\right )}{x^2 (b+c x)^2}+6 c \log (x) (2 A c-b B)+6 c (b B-2 A c) \log (b+c x)}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 138, normalized size = 1.3 \begin{align*} -{\frac{A}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{Ac}{{b}^{4}x}}-{\frac{B}{{b}^{3}x}}+6\,{\frac{{c}^{2}\ln \left ( x \right ) A}{{b}^{5}}}-3\,{\frac{Bc\ln \left ( x \right ) }{{b}^{4}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) A}{{b}^{5}}}+3\,{\frac{c\ln \left ( cx+b \right ) B}{{b}^{4}}}+3\,{\frac{A{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-2\,{\frac{Bc}{{b}^{3} \left ( cx+b \right ) }}+{\frac{A{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{Bc}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15987, size = 177, normalized size = 1.62 \begin{align*} -\frac{A b^{3} + 6 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 9 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 2 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} x}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} + \frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left (c x + b\right )}{b^{5}} - \frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52271, size = 467, normalized size = 4.28 \begin{align*} -\frac{A b^{4} + 6 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 9 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 2 \,{\left (B b^{4} - 2 \, A b^{3} c\right )} x - 6 \,{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} +{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} +{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.37443, size = 219, normalized size = 2.01 \begin{align*} - \frac{A b^{3} + x^{3} \left (- 12 A c^{3} + 6 B b c^{2}\right ) + x^{2} \left (- 18 A b c^{2} + 9 B b^{2} c\right ) + x \left (- 4 A b^{2} c + 2 B b^{3}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac{3 c \left (- 2 A c + B b\right ) \log{\left (x + \frac{- 6 A b c^{2} + 3 B b^{2} c - 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} + \frac{3 c \left (- 2 A c + B b\right ) \log{\left (x + \frac{- 6 A b c^{2} + 3 B b^{2} c + 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24284, size = 167, normalized size = 1.53 \begin{align*} -\frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{3 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{6 \, B b c^{2} x^{3} - 12 \, A c^{3} x^{3} + 9 \, B b^{2} c x^{2} - 18 \, A b c^{2} x^{2} + 2 \, B b^{3} x - 4 \, A b^{2} c x + A b^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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