Optimal. Leaf size=44 \[ \frac {c \log (c+d x)}{d (b c-a d)}-\frac {a \log (a+b x)}{b (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {72} \[ \frac {c \log (c+d x)}{d (b c-a d)}-\frac {a \log (a+b x)}{b (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {x}{(a+b x) (c+d x)} \, dx &=\int \left (-\frac {a}{(b c-a d) (a+b x)}+\frac {c}{(b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac {a \log (a+b x)}{b (b c-a d)}+\frac {c \log (c+d x)}{d (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.86 \[ -\frac {a d \log (a+b x)-b c \log (c+d x)}{b^2 c d-a b d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 38, normalized size = 0.86 \[ -\frac {a d \log \left (b x + a\right ) - b c \log \left (d x + c\right )}{b^{2} c d - a b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 46, normalized size = 1.05 \[ -\frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2} c - a b d} + \frac {c \log \left ({\left | d x + c \right |}\right )}{b c d - a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 1.02 \[ \frac {a \ln \left (b x +a \right )}{\left (a d -b c \right ) b}-\frac {c \ln \left (d x +c \right )}{\left (a d -b c \right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 44, normalized size = 1.00 \[ -\frac {a \log \left (b x + a\right )}{b^{2} c - a b d} + \frac {c \log \left (d x + c\right )}{b c d - a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 37, normalized size = 0.84 \[ \frac {a\,d\,\ln \left (a+b\,x\right )-b\,c\,\ln \left (c+d\,x\right )}{b\,d\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.83, size = 138, normalized size = 3.14 \[ \frac {a \log {\left (x + \frac {\frac {a^{3} d^{2}}{b \left (a d - b c\right )} - \frac {2 a^{2} c d}{a d - b c} + \frac {a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{b \left (a d - b c\right )} - \frac {c \log {\left (x + \frac {- \frac {a^{2} c d}{a d - b c} + \frac {2 a b c^{2}}{a d - b c} + 2 a c - \frac {b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{d \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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