Optimal. Leaf size=74 \[ -\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}-\frac{\log (a-b x)}{2 a (a d+b c)}+\frac{\log (a+b x)}{2 a (b c-a d)} \]
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Rubi [A] time = 0.0582971, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {72} \[ -\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}-\frac{\log (a-b x)}{2 a (a d+b c)}+\frac{\log (a+b x)}{2 a (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{1}{(a-b x) (a+b x) (c+d x)} \, dx &=\int \left (\frac{b}{2 a (b c+a d) (a-b x)}-\frac{b}{2 a (-b c+a d) (a+b x)}-\frac{d^2}{(b c-a d) (b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac{\log (a-b x)}{2 a (b c+a d)}+\frac{\log (a+b x)}{2 a (b c-a d)}-\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.0325434, size = 68, normalized size = 0.92 \[ \frac{(b c-a d) \log (a-b x)-(a d+b c) \log (a+b x)+2 a d \log (c+d x)}{2 a (a d-b c) (a d+b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 72, normalized size = 1. \begin{align*}{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad+bc \right ) \left ( ad-bc \right ) }}-{\frac{\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) }}-{\frac{\ln \left ( bx-a \right ) }{ \left ( 2\,ad+2\,bc \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14839, size = 96, normalized size = 1.3 \begin{align*} -\frac{d \log \left (d x + c\right )}{b^{2} c^{2} - a^{2} d^{2}} + \frac{\log \left (b x + a\right )}{2 \,{\left (a b c - a^{2} d\right )}} - \frac{\log \left (b x - a\right )}{2 \,{\left (a b c + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44592, size = 139, normalized size = 1.88 \begin{align*} -\frac{2 \, a d \log \left (d x + c\right ) -{\left (b c + a d\right )} \log \left (b x + a\right ) +{\left (b c - a d\right )} \log \left (b x - a\right )}{2 \,{\left (a b^{2} c^{2} - a^{3} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.55717, size = 668, normalized size = 9.03 \begin{align*} \frac{d \log{\left (x + \frac{\frac{12 a^{8} d^{8}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{20 a^{6} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{6 a^{6} d^{6}}{\left (a d - b c\right ) \left (a d + b c\right )} + \frac{4 a^{4} b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} + \frac{12 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right ) \left (a d + b c\right )} - 6 a^{4} d^{4} + \frac{4 a^{2} b^{6} c^{6} d^{2}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{6 a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right ) \left (a d + b c\right )} - a^{2} b^{2} c^{2} d^{2} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{\left (a d - b c\right ) \left (a d + b c\right )} - \frac{\log{\left (x + \frac{\frac{3 a^{6} d^{6}}{\left (a d + b c\right )^{2}} + \frac{3 a^{5} d^{5}}{a d + b c} - \frac{5 a^{4} b^{2} c^{2} d^{4}}{\left (a d + b c\right )^{2}} - 6 a^{4} d^{4} - \frac{6 a^{3} b^{2} c^{2} d^{3}}{a d + b c} + \frac{a^{2} b^{4} c^{4} d^{2}}{\left (a d + b c\right )^{2}} - a^{2} b^{2} c^{2} d^{2} + \frac{3 a b^{4} c^{4} d}{a d + b c} + \frac{b^{6} c^{6}}{\left (a d + b c\right )^{2}} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{2 a \left (a d + b c\right )} - \frac{\log{\left (x + \frac{\frac{3 a^{6} d^{6}}{\left (a d - b c\right )^{2}} + \frac{3 a^{5} d^{5}}{a d - b c} - \frac{5 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{2}} - 6 a^{4} d^{4} - \frac{6 a^{3} b^{2} c^{2} d^{3}}{a d - b c} + \frac{a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{2}} - a^{2} b^{2} c^{2} d^{2} + \frac{3 a b^{4} c^{4} d}{a d - b c} + \frac{b^{6} c^{6}}{\left (a d - b c\right )^{2}} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{2 a \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.9097, size = 126, normalized size = 1.7 \begin{align*} \frac{b^{2} \log \left ({\left | b x + a \right |}\right )}{2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )}} - \frac{b^{2} \log \left ({\left | b x - a \right |}\right )}{2 \,{\left (a b^{3} c + a^{2} b^{2} d\right )}} - \frac{d^{2} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - a^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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