Optimal. Leaf size=56 \[ \frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
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Rubi [A] time = 0.0327508, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=A x+B \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=A x+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b}\\ &=A x+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B (b c-a d) n \log (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.0090986, size = 56, normalized size = 1. \[ \frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 122, normalized size = 2.2 \begin{align*} Ax+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) x+{\frac{Bn{a}^{2}\ln \left ( bx+a \right ) d}{b \left ( ad-bc \right ) }}-{\frac{Bna\ln \left ( bx+a \right ) c}{ad-bc}}-{\frac{Bnc\ln \left ( dx+c \right ) a}{ad-bc}}+{\frac{Bn{c}^{2}\ln \left ( dx+c \right ) b}{d \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17749, size = 70, normalized size = 1.25 \begin{align*} B n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B x \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.841066, size = 158, normalized size = 2.82 \begin{align*} \frac{B b d n x \log \left (\frac{b x + a}{d x + c}\right ) + B a d n \log \left (b x + a\right ) - B b c n \log \left (d x + c\right ) + B b d x \log \left (e\right ) + A b d x}{b d} \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 = 1.39047, size = 72, normalized size = 1.29 \begin{align*}{\left (n x \log \left (\frac{b x + a}{d x + c}\right ) + \frac{a n \log \left (b x + a\right )}{b} - \frac{c n \log \left (-d x - c\right )}{d} + x\right )} B + A x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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