Optimal. Leaf size=86 \[ \frac{g (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d}-\frac{B g n (b c-a d)^2 \log (a+b x)}{2 b^2 d}-\frac{B g n x (b c-a d)}{2 b} \]
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Rubi [A] time = 0.0599644, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2525, 12, 43} \[ \frac{g (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d}-\frac{B g n (b c-a d)^2 \log (a+b x)}{2 b^2 d}-\frac{B g n x (b c-a d)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (c g+d g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{g (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d}-\frac{(B n) \int \frac{(b c-a d) g^2 (c+d x)}{a+b x} \, dx}{2 d g}\\ &=\frac{g (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d}-\frac{(B (b c-a d) g n) \int \frac{c+d x}{a+b x} \, dx}{2 d}\\ &=\frac{g (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d}-\frac{(B (b c-a d) g n) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{2 d}\\ &=-\frac{B (b c-a d) g n x}{2 b}-\frac{B (b c-a d)^2 g n \log (a+b x)}{2 b^2 d}+\frac{g (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0380534, size = 74, normalized size = 0.86 \[ \frac{g \left ((c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n (b c-a d) ((b c-a d) \log (a+b x)+b d x)}{b^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.328, size = 0, normalized size = 0. \begin{align*} \int \left ( dgx+cg \right ) \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 [A] time = 1.14195, size = 211, normalized size = 2.45 \begin{align*} \frac{1}{2} \, B d g x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{1}{2} \, A d g x^{2} - \frac{1}{2} \, B d g n{\left (\frac{a^{2} \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c - a d\right )} x}{b d}\right )} + B c g n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B c g x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A c g x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.804638, size = 360, normalized size = 4.19 \begin{align*} \frac{A b^{2} d^{2} g x^{2} - B b^{2} c^{2} g n \log \left (d x + c\right ) +{\left (2 \, B a b c d - B a^{2} d^{2}\right )} g n \log \left (b x + a\right ) +{\left (2 \, A b^{2} c d g -{\left (B b^{2} c d - B a b d^{2}\right )} g n\right )} x +{\left (B b^{2} d^{2} g x^{2} + 2 \, B b^{2} c d g x\right )} \log \left (e\right ) +{\left (B b^{2} d^{2} g n x^{2} + 2 \, B b^{2} c d g n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{2 \, b^{2} 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 = 3.04544, size = 173, normalized size = 2.01 \begin{align*} -\frac{B c^{2} g n \log \left (-d x - c\right )}{2 \, d} + \frac{1}{2} \,{\left (A d g + B d g\right )} x^{2} + \frac{1}{2} \,{\left (B d g n x^{2} + 2 \, B c g n x\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b c g n - B a d g n - 2 \, A b c g - 2 \, B b c g\right )} x}{2 \, b} + \frac{{\left (2 \, B a b c g n - B a^{2} d g n\right )} \log \left (b x + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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