Optimal. Leaf size=52 \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]
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Rubi [A] time = 0.11, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6686} \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6686
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{a c f+(b c+a d) f x+b d f x^2} \, dx &=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{B (b c-a d) f n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 0.96 \[ \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{f (p+1) (b B c n-a B d n)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 83, normalized size = 1.60 \[ \frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \relax (e) + A\right )} {\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \relax (e) + A\right )}^{p}}{{\left (B b c - B a d\right )} f n p + {\left (B b c - B a d\right )} f n} \]
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 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f + {\left (b c + a d\right )} f x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 14.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{p}}{b d f \,x^{2}+a c f +\left (a d +b c \right ) f x}\, dx \]
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 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f + {\left (b c + a d\right )} f x}\,{d x} \]
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 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{b\,d\,f\,x^2+f\,\left (a\,d+b\,c\right )\,x+a\,c\,f} \,d x \]
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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