Optimal. Leaf size=52 \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{B (b c-a d) \log (c+d x)}{b d}+A x \]
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Rubi [A] time = 0.027389, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{B (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=A x+B \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx\\ &=A x+\frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{(B (b c-a d)) \int \frac{1}{c+d x} \, dx}{b}\\ &=A x+\frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{B (b c-a d) \log (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.007627, size = 52, normalized size = 1. \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{B (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.149, size = 418, normalized size = 8. \begin{align*} Ax-{\frac{Ba}{b}\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{\frac{Bc}{d}\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{eBa\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}-{\frac{eBbc}{d}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}+{\frac{eB{a}^{2}d}{b \left ( dx+c \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}-2\,{\frac{eBac}{dx+c}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}+{\frac{eB{c}^{2}b}{ \left ( dx+c \right ) d}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16848, size = 73, normalized size = 1.4 \begin{align*}{\left (x \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + \frac{\frac{a e \log \left (b x + a\right )}{b} - \frac{c e \log \left (d x + c\right )}{d}}{e}\right )} B + A x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30415, size = 132, normalized size = 2.54 \begin{align*} \frac{B b d x \log \left (\frac{b e x + a e}{d x + c}\right ) + A b d x + B a d \log \left (b x + a\right ) - B b c \log \left (d x + c\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16599, size = 83, normalized size = 1.6 \begin{align*} A x + \frac{B a \log{\left (x + \frac{\frac{B a^{2} d}{b} + B a c}{B a d + B b c} \right )}}{b} - \frac{B c \log{\left (x + \frac{B a c + \frac{B b c^{2}}{d}}{B a d + B b c} \right )}}{d} + B x \log{\left (\frac{e \left (a + b x\right )}{c + d x} \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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