Optimal. Leaf size=86 \[ \frac{(b B-A c) (c d-b e)}{b^2 c (b+c x)}+\frac{\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac{\log (b+c x) (A b e-2 A c d+b B d)}{b^3}-\frac{A d}{b^2 x} \]
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Rubi [A] time = 0.0816473, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ \frac{(b B-A c) (c d-b e)}{b^2 c (b+c x)}+\frac{\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac{\log (b+c x) (A b e-2 A c d+b B d)}{b^3}-\frac{A d}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{A d}{b^2 x^2}+\frac{b B d-2 A c d+A b e}{b^3 x}+\frac{(b B-A c) (-c d+b e)}{b^2 (b+c x)^2}-\frac{c (b B d-2 A c d+A b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{A d}{b^2 x}+\frac{(b B-A c) (c d-b e)}{b^2 c (b+c x)}+\frac{(b B d-2 A c d+A b e) \log (x)}{b^3}-\frac{(b B d-2 A c d+A b e) \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.113576, size = 80, normalized size = 0.93 \[ -\frac{\frac{b (b B-A c) (b e-c d)}{c (b+c x)}-\log (x) (A b e-2 A c d+b B d)+\log (b+c x) (A b e-2 A c d+b B d)+\frac{A b d}{x}}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 133, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{Ac\ln \left ( x \right ) d}{{b}^{3}}}+{\frac{\ln \left ( x \right ) Bd}{{b}^{2}}}-{\frac{Ad}{{b}^{2}x}}+{\frac{Ae}{b \left ( cx+b \right ) }}-{\frac{Acd}{{b}^{2} \left ( cx+b \right ) }}-{\frac{Be}{c \left ( cx+b \right ) }}+{\frac{Bd}{b \left ( cx+b \right ) }}-{\frac{\ln \left ( cx+b \right ) Ae}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Acd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) Bd}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10719, size = 143, normalized size = 1.66 \begin{align*} -\frac{A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} - \frac{{\left (A b e +{\left (B b - 2 \, A c\right )} d\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (A b e +{\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.28841, size = 381, normalized size = 4.43 \begin{align*} -\frac{A b^{2} c d -{\left ({\left (B b^{2} c - 2 \, A b c^{2}\right )} d -{\left (B b^{3} - A b^{2} c\right )} e\right )} x +{\left ({\left (A b c^{2} e +{\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} +{\left (A b^{2} c e +{\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \left (c x + b\right ) -{\left ({\left (A b c^{2} e +{\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} +{\left (A b^{2} c e +{\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \left (x\right )}{b^{3} c^{2} x^{2} + b^{4} c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.71759, size = 233, normalized size = 2.71 \begin{align*} - \frac{A b c d + x \left (- A b c e + 2 A c^{2} d + B b^{2} e - B b c d\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{A b^{2} e - 2 A b c d + B b^{2} d - b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} - \frac{\left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{A b^{2} e - 2 A b c d + B b^{2} d + b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26203, size = 151, normalized size = 1.76 \begin{align*} \frac{{\left (B b d - 2 \, A c d + A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (B b c d - 2 \, A c^{2} d + A b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} + \frac{B b c d x - 2 \, A c^{2} d x - B b^{2} x e + A b c x e - A b c d}{{\left (c x^{2} + b x\right )} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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