Optimal. Leaf size=47 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f g n (b c-a d)} \]
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Rubi [A] time = 0.165762, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {6684} \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f g n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6684
Rubi steps
\begin{align*} \int \frac{1}{(a f+b f x) (c g+d g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx &=\frac{\log \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B (b c-a d) f g n}\\ \end{align*}
Mathematica [A] time = 0.123151, size = 43, normalized size = 0.91 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b B c f g n-a B d f g n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.426, size = 374, normalized size = 8. \begin{align*} -{\frac{1}{Bfgn \left ( ad-bc \right ) }\ln \left ( \ln \left ( \left ( dx+c \right ) ^{n} \right ) -{\frac{1}{2\,B} \left ( -iB\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ){\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) +iB\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ){\it csgn} \left ({\frac{i}{ \left ( dx+c \right ) ^{n}}} \right ){\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) +iB\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ) \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}+iB\pi \,{\it csgn} \left ({\frac{i}{ \left ( dx+c \right ) ^{n}}} \right ) \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \, \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}+iB\pi \,{\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \, \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}+2\,B\ln \left ( e \right ) +2\,B\ln \left ( \left ( bx+a \right ) ^{n} \right ) +2\,A \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73882, size = 72, normalized size = 1.53 \begin{align*} \frac{\log \left (-\frac{B \log \left ({\left (b x + a\right )}^{n}\right ) - B \log \left ({\left (d x + c\right )}^{n}\right ) + B \log \left (e\right ) + A}{B}\right )}{{\left (b c f g n - a d f g n\right )} B} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.513459, size = 111, normalized size = 2.36 \begin{align*} \frac{\log \left (-B n \log \left (b x + a\right ) + B n \log \left (d x + c\right ) - B \log \left (e\right ) - A\right )}{{\left (B b c - B a d\right )} f g n} \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 = 1.21471, size = 57, normalized size = 1.21 \begin{align*} \frac{\log \left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}{B b c f g n - B a d f g n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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