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 = 0.0230916, antiderivative size = 44, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0159424, 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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Maple [A] time = 0.004, size = 45, normalized size = 1. \begin{align*} -{\frac{c\ln \left ( dx+c \right ) }{d \left ( ad-bc \right ) }}+{\frac{a\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06105, size = 59, normalized size = 1.34 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19867, size = 81, normalized size = 1.84 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.726606, size = 138, normalized size = 3.14 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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