Optimal. Leaf size=1653 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.18592, antiderivative size = 1653, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {6600, 2418, 2389, 2295, 2394, 2393, 2391, 6688, 43, 2416, 6742, 2430, 2440, 2437, 2435, 6595, 6597} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6600
Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 43
Rule 2416
Rule 6742
Rule 2430
Rule 2440
Rule 2437
Rule 2435
Rule 6595
Rule 6597
Rubi steps
\begin{align*} \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x)) \, dx &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+b \int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx-(e h n) \int \left (\frac{1}{e}-\frac{d}{e (d+e x)}\right ) \text{Li}_2(c (a+b x)) \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+b \int \frac{x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx-(e h n) \int \frac{x \text{Li}_2(a c+b c x)}{d+e x} \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+b \int \left (\frac{g x \log (1-a c-b c x)}{a+b x}+\frac{h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x}\right ) \, dx-(e h n) \int \left (\frac{\text{Li}_2(a c+b c x)}{e}+\frac{d \text{Li}_2(a c+b c x)}{e (-d-e x)}\right ) \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+(b g) \int \frac{x \log (1-a c-b c x)}{a+b x} \, dx+(b h) \int \frac{x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x} \, dx-(h n) \int \text{Li}_2(a c+b c x) \, dx-(d h n) \int \frac{\text{Li}_2(a c+b c x)}{-d-e x} \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}+(b g) \int \left (\frac{\log (1-a c-b c x)}{b}-\frac{a \log (1-a c-b c x)}{b (a+b x)}\right ) \, dx+(b h) \int \left (\frac{\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{b}-\frac{a \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{b (a+b x)}\right ) \, dx-(h n) \int \log (1-a c-b c x) \, dx+(a c h n) \int \frac{\log (1-a c-b c x)}{a c+b c x} \, dx+\frac{(b c d h n) \int \frac{\log (1-a c-b c x) \log (-d-e x)}{a c+b c x} \, dx}{e}\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}+g \int \log (1-a c-b c x) \, dx-(a g) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx+h \int \log (1-a c-b c x) \log \left (f (d+e x)^n\right ) \, dx-(a h) \int \frac{\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x} \, dx+\frac{(a h n) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,a c+b c x\right )}{b}+\frac{(h n) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}+\frac{(d h n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-a b c^2-b c (1-a c)}{b c}-x\right ) \log \left (-\frac{b c d-a c e}{b c}-\frac{e x}{b c}\right )}{x} \, dx,x,a c+b c x\right )}{e}\\ &=h n x+\frac{h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac{a h n \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{(a g) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{b}-\frac{g \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}-\frac{(a h) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-a b c-b (1-a c)}{b}-c x\right ) \log \left (f \left (-\frac{-b d+a e}{b}+\frac{e x}{b}\right )^n\right )}{x} \, dx,x,a+b x\right )}{b}+(b c h) \int \frac{x \log \left (f (d+e x)^n\right )}{1-a c-b c x} \, dx-(e h n) \int \frac{x \log (1-a c-b c x)}{d+e x} \, dx\\ &=-g x+h n x-\frac{g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )+\frac{a g \text{Li}_2(c (a+b x))}{b}-\frac{a h n \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}+(b c h) \int \left (-\frac{\log \left (f (d+e x)^n\right )}{b c}+\frac{(-1+a c) \log \left (f (d+e x)^n\right )}{b c (-1+a c+b c x)}\right ) \, dx-\frac{(a h n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac{-b d+a e}{b}+\frac{e x}{b}\right )}{x} \, dx,x,a+b x\right )}{b}-(e h n) \int \left (\frac{\log (1-a c-b c x)}{e}-\frac{d \log (1-a c-b c x)}{e (d+e x)}\right ) \, dx+\frac{\left (a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{b}\\ &=-g x+h n x-\frac{g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac{a h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac{a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac{a h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac{a g \text{Li}_2(c (a+b x))}{b}-\frac{a h n \text{Li}_2(c (a+b x))}{b}-\frac{a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{a h n \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{a h n \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text{Li}_2(1-c (a+b x))}{b}+\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{a h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{a h n \text{Li}_3(1-c (a+b x))}{b}+\frac{a h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-h \int \log \left (f (d+e x)^n\right ) \, dx-((1-a c) h) \int \frac{\log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx-(h n) \int \log (1-a c-b c x) \, dx+(d h n) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx\\ &=-g x+h n x-\frac{g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac{(1-a c) h \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac{a h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac{a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac{a h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac{a g \text{Li}_2(c (a+b x))}{b}-\frac{a h n \text{Li}_2(c (a+b x))}{b}-\frac{a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{a h n \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{a h n \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text{Li}_2(1-c (a+b x))}{b}+\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{a h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{a h n \text{Li}_3(1-c (a+b x))}{b}+\frac{a h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac{h \operatorname{Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{(h n) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}+\frac{(b c d h n) \int \frac{\log \left (-\frac{b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{e}+\frac{((1-a c) e h n) \int \frac{\log \left (\frac{e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{b c}\\ &=-g x+3 h n x-\frac{g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{2 h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}-\frac{h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac{(1-a c) h \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac{a h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac{a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac{a h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac{a g \text{Li}_2(c (a+b x))}{b}-\frac{a h n \text{Li}_2(c (a+b x))}{b}-\frac{a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{a h n \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{a h n \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text{Li}_2(1-c (a+b x))}{b}+\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{a h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{a h n \text{Li}_3(1-c (a+b x))}{b}+\frac{a h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}+\frac{((1-a c) h n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{b c}-\frac{(d h n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{e}\\ &=-g x+3 h n x-\frac{g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{2 h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac{d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac{d h n \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac{d h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac{d h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}-\frac{h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac{(1-a c) h \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac{a h n \left (\log (c (a+b x))+\log \left (\frac{b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac{(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac{a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac{a h n \left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac{a g \text{Li}_2(c (a+b x))}{b}-\frac{a h n \text{Li}_2(c (a+b x))}{b}-\frac{a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d h n \left (\log (-d-e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2(1-a c-b c x)}{e}+\frac{d h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{e}-h n x \text{Li}_2(a c+b c x)+\frac{d h n \log (-d-e x) \text{Li}_2(a c+b c x)}{e}-\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{d h n \left (\log (1-a c-b c x)+\log \left (\frac{b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{e}-\frac{a h n \left (\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text{Li}_2\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{(1-a c) h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{b c}-\frac{a h n \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text{Li}_2(1-c (a+b x))}{b}+\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \log \left (\frac{b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text{Li}_2\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac{d h n \text{Li}_3(1-a c-b c x)}{e}-\frac{d h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac{d h n \text{Li}_3\left (\frac{(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac{a h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{b}-\frac{d h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{a h n \text{Li}_3(1-c (a+b x))}{b}+\frac{a h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac{a h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}\\ \end{align*}
Mathematica [A] time = 4.92197, size = 1546, normalized size = 0.94 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.285, size = 0, normalized size = 0. \begin{align*} \int \left ( g+h\ln \left ( f \left ( ex+d \right ) ^{n} \right ) \right ){\it polylog} \left ( 2,c \left ( bx+a \right ) \right ) \, 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*} \frac{{\left (d h n \log \left (e x + d\right ) + e h x \log \left ({\left (e x + d\right )}^{n}\right ) -{\left (e h n - e h \log \left (f\right ) - e g\right )} x\right )}{\rm Li}_2\left (b c x + a c\right )}{e} + \int \frac{b e h x \log \left (-b c x - a c + 1\right ) \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b d h n \log \left (e x + d\right ) -{\left (b e h n - b e h \log \left (f\right ) - b e g\right )} x\right )} \log \left (-b c x - a c + 1\right )}{b e x + a e}\,{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 (h{\rm Li}_2\left (b c x + a c\right ) \log \left ({\left (e x + d\right )}^{n} f\right ) + g{\rm Li}_2\left (b c x + a c\right ), 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{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )}{\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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