Optimal. Leaf size=41 \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2505} \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2505
Rubi steps
\begin {align*} \int \frac {\log ^p\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a c+(b c+a d) x+b d x^2} \, dx &=\frac {\log ^{1+p}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.98 \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(p+1) (b c n-a d n)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 65, normalized size = 1.59 \[ \frac {{\left (n \log \left (\frac {b x + a}{d x + c}\right ) + \log \relax (e)\right )} {\left (n \log \left (\frac {b x + a}{d x + c}\right ) + \log \relax (e)\right )}^{p}}{{\left (b c - a d\right )} n p + {\left (b c - a d\right )} n} \]
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 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{p}}{b d x^{2} + a c + {\left (b c + a d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{p}}{b d \,x^{2}+a c +\left (a d +b c \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{p}}{b d x^{2} + a c + {\left (b c + a d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^p}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c} \,d x \]
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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