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 = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.95, size = 64, normalized size = 0.86 \[ -\frac {2 \, a d \log \left (d x + c\right ) - {\left (b c + a d\right )} \log \left (b x + a\right ) + {\left (b c - a d\right )} \log \left (b x - a\right )}{2 \, {\left (a b^{2} c^{2} - a^{3} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 93, normalized size = 1.26 \[ \frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )}} - \frac {b^{2} \log \left ({\left | b x - a \right |}\right )}{2 \, {\left (a b^{3} c + a^{2} b^{2} d\right )}} - \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - a^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.97 \[ \frac {d \ln \left (d x +c \right )}{\left (a d +b c \right ) \left (a d -b c \right )}-\frac {\ln \left (b x -a \right )}{2 \left (a d +b c \right ) a}-\frac {\ln \left (b x +a \right )}{2 \left (a d -b c \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 71, normalized size = 0.96 \[ -\frac {d \log \left (d x + c\right )}{b^{2} c^{2} - a^{2} d^{2}} + \frac {\log \left (b x + a\right )}{2 \, {\left (a b c - a^{2} d\right )}} - \frac {\log \left (b x - a\right )}{2 \, {\left (a b c + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 72, normalized size = 0.97 \[ \frac {d\,\ln \left (c+d\,x\right )}{a^2\,d^2-b^2\,c^2}-\frac {\ln \left (a-b\,x\right )}{2\,d\,a^2+2\,b\,c\,a}-\frac {\ln \left (a+b\,x\right )}{2\,a^2\,d-2\,a\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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