Optimal. Leaf size=44 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]
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Rubi [A] time = 0.0775872, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {6684} \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6684
Rubi steps
\begin{align*} \int \frac{1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \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 n}\\ \end{align*}
Mathematica [A] time = 0.0544395, size = 42, normalized size = 0.95 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{f (b B c n-a B d n)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.418, size = 371, normalized size = 8.4 \begin{align*} -{\frac{1}{Bfn \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 = 1.78736, size = 69, normalized size = 1.57 \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 n - a d f n\right )} B} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487774, size = 108, normalized size = 2.45 \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 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 = 1.22325, size = 54, normalized size = 1.23 \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 n - B a d f n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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