Optimal. Leaf size=41 \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n (b c-a d)} \]
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Rubi [A] time = 0.12, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6684} \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6684
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d 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) n}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 39, normalized size = 0.95 \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b B c n-a B d n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 45, normalized size = 1.10 \[ \frac {\log \left (-B n \log \left (b x + a\right ) + B n \log \left (d x + c\right ) - B \log \relax (e) - A\right )}{{\left (B b c - B a d\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 38, normalized size = 0.93 \[ \frac {\log \left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}{B b c n - B a d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 368, normalized size = 8.98 \[ -\frac {\ln \left (\ln \left (\left (d x +c \right )^{n}\right )-\frac {-i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i \pi B \,\mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+2 B \ln \relax (e )+2 B \ln \left (\left (b x +a \right )^{n}\right )+2 A}{2 B}\right )}{\left (a d -b c \right ) B n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 49, normalized size = 1.20 \[ \frac {\log \left (-\frac {B \log \left ({\left (b x + a\right )}^{n}\right ) - B \log \left ({\left (d x + c\right )}^{n}\right ) + B \log \relax (e) + A}{B}\right )}{{\left (b c n - a d n\right )} B} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 40, normalized size = 0.98 \[ -\frac {\ln \left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}{B\,a\,d\,n-B\,b\,c\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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