Optimal. Leaf size=54 \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]
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Rubi [A] time = 0.0271851, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]
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)^2}{(c+d x)^2}\right )\right ) \, dx &=A x+B \int \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right ) \, dx\\ &=A x+\frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{(2 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)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.0232233, size = 54, normalized size = 1. \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.224, size = 233, normalized size = 4.3 \begin{align*} Ax+B\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) x+{\frac{Bc}{d}\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) }-2\,{\frac{B\ln \left ( \left ( dx+c \right ) ^{-1} \right ) a}{b}}+2\,{\frac{B\ln \left ( \left ( dx+c \right ) ^{-1} \right ) c}{d}}+2\,{\frac{Bd{a}^{2}}{b \left ( ad-bc \right ) }\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) }-4\,{\frac{Bac}{ad-bc}\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) }+2\,{\frac{B{c}^{2}b}{d \left ( ad-bc \right ) }\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23, size = 77, normalized size = 1.43 \begin{align*}{\left (x \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + \frac{2 \,{\left (\frac{a e \log \left (b x + a\right )}{b} - \frac{c e \log \left (d x + c\right )}{d}\right )}}{e}\right )} B + A x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01892, size = 184, normalized size = 3.41 \begin{align*} \frac{B b d x \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + A b d x + 2 \, B a d \log \left (b x + a\right ) - 2 \, 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 [B] time = 1.14493, size = 104, normalized size = 1.93 \begin{align*} A x + \frac{2 B a \log{\left (x + \frac{\frac{2 B a^{2} d}{b} + 2 B a c}{2 B a d + 2 B b c} \right )}}{b} - \frac{2 B c \log{\left (x + \frac{2 B a c + \frac{2 B b c^{2}}{d}}{2 B a d + 2 B b c} \right )}}{d} + B x \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \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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