Optimal. Leaf size=67 \[ -\frac{(c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac{B n}{b g^2 (a+b x)} \]
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Rubi [A] time = 0.0904321, antiderivative size = 108, normalized size of antiderivative = 1.61, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{b g^2 (a+b x)}-\frac{B d n \log (a+b x)}{b g^2 (b c-a d)}+\frac{B d n \log (c+d x)}{b g^2 (b c-a d)}-\frac{B n}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b g^2 (a+b x)}+\frac{(B n) \int \frac{b c-a d}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b g^2 (a+b x)}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b g^2 (a+b x)}+\frac{(B (b c-a d) n) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac{B n}{b g^2 (a+b x)}-\frac{B d n \log (a+b x)}{b (b c-a d) g^2}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b g^2 (a+b x)}+\frac{B d n \log (c+d x)}{b (b c-a d) g^2}\\ \end{align*}
Mathematica [A] time = 0.0633145, size = 115, normalized size = 1.72 \[ \frac{B n (b c-a d) \left (-\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2}\right )}{b g^2}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{b g (a g+b g x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21445, size = 185, normalized size = 2.76 \begin{align*} -B n{\left (\frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac{B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac{A}{b^{2} g^{2} x + a b g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.796695, size = 221, normalized size = 3.3 \begin{align*} -\frac{A b c - A a d +{\left (B b c - B a d\right )} n +{\left (B b c - B a d\right )} \log \left (e\right ) +{\left (B b d n x + B b c n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x +{\left (a b^{2} c - a^{2} b d\right )} g^{2}} \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 = 1.42295, size = 162, normalized size = 2.42 \begin{align*} -\frac{B d n \log \left (b x + a\right )}{b^{2} c g^{2} - a b d g^{2}} + \frac{B d n \log \left (d x + c\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac{B n \log \left (\frac{b x + a}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac{B n + A + B}{b^{2} g^{2} x + a b g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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