Optimal. Leaf size=45 \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 B n (b c-a d)} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6686} \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 B n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6686
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx &=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 B (b c-a d) n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 0.96 \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (b B c n-a B d n)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 72, normalized size = 1.60 \[ \frac {B n \log \left (b x + a\right )^{2} + B n \log \left (d x + c\right )^{2} + 2 \, {\left (B \log \relax (e) + A\right )} \log \left (b x + a\right ) - 2 \, {\left (B n \log \left (b x + a\right ) + B \log \relax (e) + A\right )} \log \left (d x + c\right )}{2 \, {\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.61, size = 1152, normalized size = 25.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.20, size = 151, normalized size = 3.36 \[ B {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} - \frac {{\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} B}{2 \, {\left (b c - a d\right )} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 71, normalized size = 1.58 \[ -\frac {B\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2-A\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{2\,n\,\left (a\,d-b\,c\right )} \]
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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