Optimal. Leaf size=240 \[ \frac{2 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f^3}-\frac{2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^3}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}+\frac{2 g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{b e g n \log (d+e x)}{f^2 (e f-d g)}-\frac{b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2} \]
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Rubi [A] time = 0.244097, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac{2 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f^3}-\frac{2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^3}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}+\frac{2 g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{b e g n \log (d+e x)}{f^2 (e f-d g)}-\frac{b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2416
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2 (f+g x)^2} \, dx &=\int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x^2}-\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)^2}+\frac{2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^2}-\frac{(2 g) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^3}+\frac{\left (2 g^2\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f^3}+\frac{g^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{f^2}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f^3}+\frac{(b e n) \int \frac{1}{x (d+e x)} \, dx}{f^2}+\frac{(2 b e g n) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{f^3}-\frac{(2 b e g n) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f^3}+\frac{(b e g n) \int \frac{1}{(d+e x) (f+g x)} \, dx}{f^2}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f^3}-\frac{2 b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}+\frac{(b e n) \int \frac{1}{x} \, dx}{d f^2}-\frac{\left (b e^2 n\right ) \int \frac{1}{d+e x} \, dx}{d f^2}-\frac{(2 b g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f^3}+\frac{\left (b e^2 g n\right ) \int \frac{1}{d+e x} \, dx}{f^2 (e f-d g)}-\frac{\left (b e g^2 n\right ) \int \frac{1}{f+g x} \, dx}{f^2 (e f-d g)}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}+\frac{b e g n \log (d+e x)}{f^2 (e f-d g)}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f^3}+\frac{2 b g n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{f^3}-\frac{2 b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^3}\\ \end{align*}
Mathematica [A] time = 0.208285, size = 199, normalized size = 0.83 \[ \frac{2 b g n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-\frac{f g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+2 g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e f g n (\log (d+e x)-\log (f+g x))}{e f-d g}+\frac{b e f n (\log (x)-\log (d+e x))}{d}}{f^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.519, size = 936, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{2 \, g x + f}{f^{2} g x^{2} + f^{3} x} - \frac{2 \, g \log \left (g x + f\right )}{f^{3}} + \frac{2 \, g \log \left (x\right )}{f^{3}}\right )} + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g^{2} x^{4} + 2 \, f g x^{3} + f^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g^{2} x^{4} + 2 \, f g x^{3} + f^{2} x^{2}}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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