Optimal. Leaf size=427 \[ \frac{2 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}-\frac{2 b g n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{2 b^2 e n^2 \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}-\frac{2 b^2 g n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}+\frac{2 b^2 g n^2 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac{2 b e n \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+i x) (e h-d i) (g h-f i)}+\frac{g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{g \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2} \]
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Rubi [A] time = 0.490768, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2418, 2396, 2433, 2374, 6589, 2397, 2394, 2393, 2391} \[ \frac{2 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}-\frac{2 b g n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{2 b^2 e n^2 \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}-\frac{2 b^2 g n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}+\frac{2 b^2 g n^2 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac{2 b e n \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+i x) (e h-d i) (g h-f i)}+\frac{g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{g \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2397
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+228 x)^2 (f+g x)} \, dx &=\int \left (\frac{228 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h) (h+228 x)^2}-\frac{228 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2 (h+228 x)}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2 (f+g x)}\right ) \, dx\\ &=-\frac{(228 g) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{h+228 x} \, dx}{(228 f-g h)^2}+\frac{g^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx}{(228 f-g h)^2}+\frac{228 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+228 x)^2} \, dx}{228 f-g h}\\ &=-\frac{228 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 d-e h) (228 f-g h) (h+228 x)}-\frac{g \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{(2 b e g n) \int \frac{\log \left (\frac{e (h+228 x)}{-228 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{(228 f-g h)^2}-\frac{(2 b e g n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{(228 f-g h)^2}+\frac{(456 b e n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{h+228 x} \, dx}{(228 d-e h) (228 f-g h)}\\ &=\frac{2 b e n \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(228 d-e h) (228 f-g h)}-\frac{228 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 d-e h) (228 f-g h) (h+228 x)}-\frac{g \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{(2 b g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{-228 d+e h}{e}+\frac{228 x}{e}\right )}{-228 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(228 f-g h)^2}-\frac{(2 b g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(228 f-g h)^2}-\frac{\left (2 b^2 e^2 n^2\right ) \int \frac{\log \left (\frac{e (h+228 x)}{-228 d+e h}\right )}{d+e x} \, dx}{(228 d-e h) (228 f-g h)}\\ &=\frac{2 b e n \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(228 d-e h) (228 f-g h)}-\frac{228 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 d-e h) (228 f-g h) (h+228 x)}-\frac{g \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(228 f-g h)^2}-\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{228 (d+e x)}{228 d-e h}\right )}{(228 f-g h)^2}-\frac{\left (2 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(228 f-g h)^2}+\frac{\left (2 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{228 x}{-228 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(228 f-g h)^2}-\frac{\left (2 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{228 x}{-228 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(228 d-e h) (228 f-g h)}\\ &=\frac{2 b e n \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(228 d-e h) (228 f-g h)}-\frac{228 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 d-e h) (228 f-g h) (h+228 x)}-\frac{g \log \left (-\frac{e (h+228 x)}{228 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(228 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{2 b^2 e n^2 \text{Li}_2\left (\frac{228 (d+e x)}{228 d-e h}\right )}{(228 d-e h) (228 f-g h)}-\frac{2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{228 (d+e x)}{228 d-e h}\right )}{(228 f-g h)^2}-\frac{2 b^2 g n^2 \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{(228 f-g h)^2}+\frac{2 b^2 g n^2 \text{Li}_3\left (\frac{228 (d+e x)}{228 d-e h}\right )}{(228 f-g h)^2}\\ \end{align*}
Mathematica [A] time = 0.793132, size = 630, normalized size = 1.48 \[ \frac{-2 b n \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right ) \left (-g (h+i x) (e h-d i) \left (\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )+g (h+i x) (e h-d i) \left (\text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )+\log (d+e x) \log \left (\frac{e (h+i x)}{e h-d i}\right )\right )+(g h-f i) (i (d+e x) \log (d+e x)-e (h+i x) \log (h+i x))\right )-b^2 n^2 \left (-g (h+i x) (e h-d i) \left (-2 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\log ^2(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )+(g h-f i) \left (\log (d+e x) \left (i (d+e x) \log (d+e x)-2 e (h+i x) \log \left (\frac{e (h+i x)}{e h-d i}\right )\right )-2 e (h+i x) \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )\right )+g (h+i x) (e h-d i) \left (-2 \text{PolyLog}\left (3,\frac{i (d+e x)}{d i-e h}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )+\log ^2(d+e x) \log \left (\frac{e (h+i x)}{e h-d i}\right )\right )\right )+(e h-d i) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+g (h+i x) (e h-d i) \log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-g (h+i x) (e h-d i) \log (h+i x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{(h+i x) (e h-d i) (g h-f i)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.826, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{ \left ( gx+f \right ) \left ( ix+h \right ) ^{2}}}\, dx \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^{2}{\left (\frac{g \log \left (g x + f\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} - \frac{g \log \left (i x + h\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} + \frac{1}{g h^{2} - f h i +{\left (g h i - f i^{2}\right )} x}\right )} + \int \frac{b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g i^{2} x^{3} + f h^{2} +{\left (2 \, g h i + f i^{2}\right )} x^{2} +{\left (g h^{2} + 2 \, f h i\right )} x}\,{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^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g i^{2} x^{3} + f h^{2} +{\left (2 \, g h i + f i^{2}\right )} x^{2} +{\left (g h^{2} + 2 \, f h i\right )} x}, 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{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}{\left (i x + h\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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