Optimal. Leaf size=57 \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
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Rubi [A] time = 0.0299171, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=A x+B \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=A x+\frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b}\\ &=A x-\frac{B (b c-a d) n \log (c+d x)}{b d}+\frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0107576, size = 57, normalized size = 1. \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 123, normalized size = 2.2 \begin{align*} Ax+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) x+{\frac{Bn{a}^{2}\ln \left ( bx+a \right ) d}{b \left ( ad-bc \right ) }}-{\frac{Bna\ln \left ( bx+a \right ) c}{ad-bc}}-{\frac{Bnc\ln \left ( dx+c \right ) a}{ad-bc}}+{\frac{Bn{c}^{2}\ln \left ( dx+c \right ) b}{d \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12728, size = 80, normalized size = 1.4 \begin{align*} B x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A x + \frac{{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} B}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03051, size = 146, normalized size = 2.56 \begin{align*} \frac{B b d x \log \left (e\right ) + A b d x +{\left (B b d n x + B a d n\right )} \log \left (b x + a\right ) -{\left (B b d n x + B b c n\right )} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32551, size = 74, normalized size = 1.3 \begin{align*}{\left (n x \log \left (b x + a\right ) - n x \log \left (d x + c\right ) + \frac{a n \log \left (b x + a\right )}{b} - \frac{c n \log \left (-d x - c\right )}{d} + x\right )} B + A x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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