Optimal. Leaf size=335 \[ -\frac{3 b g^2 n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f^4}+\frac{3 b g^2 n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^4}+\frac{3 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}-\frac{3 g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}-\frac{b e^2 n \log (x)}{2 d^2 f^2}+\frac{b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac{b e g^2 n \log (d+e x)}{f^3 (e f-d g)}+\frac{b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac{2 b e g n \log (x)}{d f^3}+\frac{2 b e g n \log (d+e x)}{d f^3}-\frac{b e n}{2 d f^2 x} \]
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Rubi [A] time = 0.314122, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{3 b g^2 n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f^4}+\frac{3 b g^2 n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^4}+\frac{3 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}-\frac{3 g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}-\frac{b e^2 n \log (x)}{2 d^2 f^2}+\frac{b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac{b e g^2 n \log (d+e x)}{f^3 (e f-d g)}+\frac{b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac{2 b e g n \log (x)}{d f^3}+\frac{2 b e g n \log (d+e x)}{d f^3}-\frac{b e n}{2 d f^2 x} \]
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^3 (f+g x)^2} \, dx &=\int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x^3}-\frac{2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x^2}+\frac{3 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4 x}-\frac{g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)^2}-\frac{3 g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4 (f+g x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx}{f^2}-\frac{(2 g) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^3}+\frac{\left (3 g^2\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^4}-\frac{\left (3 g^3\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f^4}-\frac{g^3 \int \frac{a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{f^3}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac{3 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}-\frac{3 g^2 \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^4}+\frac{(b e n) \int \frac{1}{x^2 (d+e x)} \, dx}{2 f^2}-\frac{(2 b e g n) \int \frac{1}{x (d+e x)} \, dx}{f^3}-\frac{\left (3 b e g^2 n\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{f^4}+\frac{\left (3 b e g^2 n\right ) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f^4}-\frac{\left (b e g^2 n\right ) \int \frac{1}{(d+e x) (f+g x)} \, dx}{f^3}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac{3 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}-\frac{3 g^2 \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^4}+\frac{3 b g^2 n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^4}+\frac{(b e n) \int \left (\frac{1}{d x^2}-\frac{e}{d^2 x}+\frac{e^2}{d^2 (d+e x)}\right ) \, dx}{2 f^2}-\frac{(2 b e g n) \int \frac{1}{x} \, dx}{d f^3}+\frac{\left (2 b e^2 g n\right ) \int \frac{1}{d+e x} \, dx}{d f^3}+\frac{\left (3 b g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f^4}-\frac{\left (b e^2 g^2 n\right ) \int \frac{1}{d+e x} \, dx}{f^3 (e f-d g)}+\frac{\left (b e g^3 n\right ) \int \frac{1}{f+g x} \, dx}{f^3 (e f-d g)}\\ &=-\frac{b e n}{2 d f^2 x}-\frac{b e^2 n \log (x)}{2 d^2 f^2}-\frac{2 b e g n \log (x)}{d f^3}+\frac{b e^2 n \log (d+e x)}{2 d^2 f^2}+\frac{2 b e g n \log (d+e x)}{d f^3}-\frac{b e g^2 n \log (d+e x)}{f^3 (e f-d g)}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac{3 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac{b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac{3 g^2 \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^4}-\frac{3 b g^2 n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{f^4}+\frac{3 b g^2 n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^4}\\ \end{align*}
Mathematica [A] time = 0.399546, size = 269, normalized size = 0.8 \[ -\frac{6 b g^2 n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-6 b g^2 n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2}-\frac{2 f g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+6 g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{4 f g \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-6 g^2 \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e f^2 n (-e x \log (d+e x)+d+e x \log (x))}{d^2 x}+\frac{2 b e f g^2 n (\log (d+e x)-\log (f+g x))}{e f-d g}+\frac{4 b e f g n (\log (x)-\log (d+e x))}{d}}{2 f^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.536, size = 1224, normalized size = 3.7 \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*} \frac{1}{2} \, a{\left (\frac{6 \, g^{2} x^{2} + 3 \, f g x - f^{2}}{f^{3} g x^{3} + f^{4} x^{2}} - \frac{6 \, g^{2} \log \left (g x + f\right )}{f^{4}} + \frac{6 \, g^{2} \log \left (x\right )}{f^{4}}\right )} + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g^{2} x^{5} + 2 \, f g x^{4} + f^{2} x^{3}}\,{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^{5} + 2 \, f g x^{4} + f^{2} x^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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