Optimal. Leaf size=3733 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 4.30785, antiderivative size = 3733, normalized size of antiderivative = 1., number of steps used = 78, number of rules used = 18, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6603, 2439, 44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391, 2438, 2437, 2435, 2440, 6598, 6597} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6603
Rule 2439
Rule 44
Rule 2416
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rule 2438
Rule 2437
Rule 2435
Rule 2440
Rule 6598
Rule 6597
Rubi steps
\begin{align*} \int \frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{x^4} \, dx &=-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{1}{3} b \int \left (\frac{\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a x^3}-\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^2 x^2}+\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^3 x}-\frac{b^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^3 (a+b x)}\right ) \, dx+\frac{1}{3} (e h n) \int \left (\frac{\text{Li}_2(c (a+b x))}{d x^3}-\frac{e \text{Li}_2(c (a+b x))}{d^2 x^2}+\frac{e^2 \text{Li}_2(c (a+b x))}{d^3 x}-\frac{e^3 \text{Li}_2(c (a+b x))}{d^3 (d+e x)}\right ) \, dx\\ &=-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{b \int \frac{\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x^3} \, dx}{3 a}+\frac{b^2 \int \frac{\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x^2} \, dx}{3 a^2}-\frac{b^3 \int \frac{\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x} \, dx}{3 a^3}+\frac{b^4 \int \frac{\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx}{3 a^3}+\frac{(e h n) \int \frac{\text{Li}_2(c (a+b x))}{x^3} \, dx}{3 d}-\frac{\left (e^2 h n\right ) \int \frac{\text{Li}_2(c (a+b x))}{x^2} \, dx}{3 d^2}+\frac{\left (e^3 h n\right ) \int \frac{\text{Li}_2(c (a+b x))}{x} \, dx}{3 d^3}-\frac{\left (e^4 h n\right ) \int \frac{\text{Li}_2(c (a+b x))}{d+e x} \, dx}{3 d^3}\\ &=\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-a b c-b (1-a c)}{b}-c x\right ) \left (g+h \log \left (f \left (-\frac{-b d+a e}{b}+\frac{e x}{b}\right )^n\right )\right )}{x} \, dx,x,a+b x\right )}{3 a^3}+\frac{\left (b^2 c\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{x^2 (1-a c-b c x)} \, dx}{6 a}-\frac{\left (b^3 c\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{x (1-a c-b c x)} \, dx}{3 a^2}-\frac{\left (b^3 g\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{3 a^3}-\frac{\left (b^3 h\right ) \int \frac{\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{x} \, dx}{3 a^3}-\frac{(b e h n) \int \frac{\log (1-a c-b c x)}{x^2 (d+e x)} \, dx}{6 a}+\frac{\left (b^2 e h n\right ) \int \frac{\log (1-a c-b c x)}{x (d+e x)} \, dx}{3 a^2}-\frac{(b e h n) \int \frac{\log (1-a c-b c x)}{x^2 (a+b x)} \, dx}{6 d}+\frac{\left (b e^2 h n\right ) \int \frac{\log (1-a c-b c x)}{x (a+b x)} \, dx}{3 d^2}+\frac{\left (b e^3 h n\right ) \int \frac{\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{3 d^3}-\frac{\left (b e^3 h n\right ) \int \frac{\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{3 d^3}\\ &=-\frac{b^3 g \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}+\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}+\frac{\left (b^2 c\right ) \int \left (-\frac{g+h \log \left (f (d+e x)^n\right )}{(-1+a c) x^2}+\frac{b c \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c)^2 x}-\frac{b^2 c^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c)^2 (-1+a c+b c x)}\right ) \, dx}{6 a}-\frac{\left (b^3 c\right ) \int \left (-\frac{g+h \log \left (f (d+e x)^n\right )}{(-1+a c) x}+\frac{b c \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c) (-1+a c+b c x)}\right ) \, dx}{3 a^2}+\frac{\left (b^3 g\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 )}{3 a^3}-\frac{\left (b^4 c g\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a^3}+\frac{\left (b^3 h\right ) \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 )}{3 a^3}-\frac{\left (b^3 h n\right ) \int \frac{\log (1-a c-b c x) \log (d+e x)}{x} \, dx}{3 a^3}-\frac{(b e h n) \int \left (\frac{\log (1-a c-b c x)}{d x^2}-\frac{e \log (1-a c-b c x)}{d^2 x}+\frac{e^2 \log (1-a c-b c x)}{d^2 (d+e x)}\right ) \, dx}{6 a}+\frac{\left (b^2 e h n\right ) \int \left (\frac{\log (1-a c-b c x)}{d x}-\frac{e \log (1-a c-b c x)}{d (d+e x)}\right ) \, dx}{3 a^2}-\frac{(b e h n) \int \left (\frac{\log (1-a c-b c x)}{a x^2}-\frac{b \log (1-a c-b c x)}{a^2 x}+\frac{b^2 \log (1-a c-b c x)}{a^2 (a+b x)}\right ) \, dx}{6 d}+\frac{\left (b e^2 h n\right ) \int \left (\frac{\log (1-a c-b c x)}{a x}-\frac{b \log (1-a c-b c x)}{a (a+b x)}\right ) \, dx}{3 d^2}+\frac{\left (e^3 h n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{a}{b}+\frac{x}{b}\right ) \log \left (-\frac{-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{3 d^3}-\frac{\left (e^3 h n\right ) \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 )}{3 d^3}+\frac{\left (b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{3 a^3}\\ &=-\frac{b^3 g \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}-\frac{b^3 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{3 a^3}-\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )+\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) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{6 a^3}+\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )-\log \left (-\frac{e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{6 a^3}+\frac{b^3 h \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{3 a^3}+\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{e^3 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 )}{6 d^3}+\frac{e^3 h n \log (x) \log \left (1+\frac{b x}{a}\right ) \log (1-c (a+b x))}{3 d^3}-\frac{e^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 d^3}+\frac{e^3 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}{6 d^3}+\frac{e^3 h n \left (\log \left (1+\frac{b x}{a}\right )+\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )}{6 d^3}+\frac{e^3 h n \left (\log (c (a+b x))-\log \left (1+\frac{b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )^2}{6 d^3}+\frac{e^3 h n \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{3 d^3}-\frac{b^3 g \text{Li}_2(c (a+b x))}{3 a^3}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{b^3 g \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \left (\log (d+e x)-\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{b^3 h n \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}-\frac{e^3 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))}{3 d^3}+\frac{e^3 h n \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2(1-c (a+b x))}{3 d^3}+\frac{e^3 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 )}{3 d^3}-\frac{e^3 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 )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 d^3}+\frac{\left (b^3 c^2\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{x} \, dx}{6 a (1-a c)^2}-\frac{\left (b^4 c^3\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{6 a (1-a c)^2}+\frac{\left (b^2 c\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{x^2} \, dx}{6 a (1-a c)}-\frac{\left (b^3 c\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{x} \, dx}{3 a^2 (1-a c)}+\frac{\left (b^4 c^2\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{3 a^2 (1-a c)}+\frac{\left (b^3 h n\right ) \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 )}{3 a^3}-2 \frac{(b e h n) \int \frac{\log (1-a c-b c x)}{x^2} \, dx}{6 a d}+\frac{\left (b^2 e h n\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{6 a^2 d}+\frac{\left (b^2 e h n\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{3 a^2 d}-\frac{\left (b^3 e h n\right ) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{6 a^2 d}+\frac{\left (b e^2 h n\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{6 a d^2}+\frac{\left (b e^2 h n\right ) \int \frac{\log (1-a c-b c x)}{x} \, dx}{3 a d^2}-\frac{\left (b^2 e^2 h n\right ) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{3 a d^2}-\frac{\left (b^2 e^2 h n\right ) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{3 a^2 d}-\frac{\left (b e^3 h n\right ) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{6 a d^2}-\frac{\left (b^3 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 )}{3 a^3}+\frac{\left (b^4 c h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a^3}\\ &=-\frac{b^3 g \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}+\frac{b^2 e h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2 d}+\frac{b e^2 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a d^2}-\frac{b^3 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{3 a^3}-\frac{b^2 e 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 )}{3 a^2 d}-\frac{b e^2 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 )}{6 a d^2}-\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )+\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) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{6 a^3}+\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )-\log \left (-\frac{e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{6 a^3}+\frac{b^3 h \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{3 a^3}-\frac{b^2 c \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c) x}+\frac{b^3 c^2 \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}-\frac{b^3 c \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{b^3 c^2 \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}+\frac{b^3 c \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b^3 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 )}{6 a^3}-\frac{e^3 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 )}{6 d^3}+\frac{e^3 h n \log (x) \log \left (1+\frac{b x}{a}\right ) \log (1-c (a+b x))}{3 d^3}+\frac{b^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 a^3}-\frac{e^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 d^3}-\frac{b^3 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}{6 a^3}+\frac{e^3 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}{6 d^3}+\frac{e^3 h n \left (\log \left (1+\frac{b x}{a}\right )+\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )}{6 d^3}+\frac{e^3 h n \left (\log (c (a+b x))-\log \left (1+\frac{b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )^2}{6 d^3}+\frac{e^3 h n \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{3 d^3}-\frac{b^3 g \text{Li}_2(c (a+b x))}{3 a^3}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 a^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{b^3 g \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \left (\log (d+e x)-\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (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 )}{3 a^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{b^3 h n \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}+\frac{b^3 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))}{3 a^3}-\frac{e^3 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))}{3 d^3}+\frac{e^3 h n \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2(1-c (a+b x))}{3 d^3}-\frac{b^3 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 )}{3 a^3}+\frac{e^3 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 )}{3 d^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}-\frac{b^3 h n \text{Li}_3(1-c (a+b x))}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 a^3}-\frac{e^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 d^3}-\frac{\left (b^3 c^2 e h n\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{6 a (1-a c)^2}+\frac{\left (b^3 c^2 e h n\right ) \int \frac{\log \left (\frac{e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{6 a (1-a c)^2}+\frac{\left (b^2 c e h n\right ) \int \frac{1}{x (d+e x)} \, dx}{6 a (1-a c)}+\frac{\left (b^3 c e h n\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{3 a^2 (1-a c)}-\frac{\left (b^3 c e h n\right ) \int \frac{\log \left (\frac{e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{3 a^2 (1-a c)}-\frac{\left (b^2 e h n\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{6 a^2 d}-2 \left (-\frac{b e h n \log (1-a c-b c x)}{6 a d x}-\frac{\left (b^2 c e h n\right ) \int \frac{1}{x (1-a c-b c x)} \, dx}{6 a d}\right )+\frac{\left (b^3 c e h n\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{6 a^2 d}+\frac{\left (b^3 c e h n\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a^2 d}-\frac{\left (b^3 c e h n\right ) \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}{3 a^2 d}-\frac{\left (b e^2 h n\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 a d^2}+\frac{\left (b^2 c e^2 h n\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{6 a d^2}-\frac{\left (b^2 c e^2 h n\right ) \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}{6 a d^2}+\frac{\left (b^2 c e^2 h n\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a d^2}\\ &=-\frac{b^3 g \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}+\frac{b^2 e h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2 d}+\frac{b e^2 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a d^2}-\frac{b^3 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{3 a^3}-\frac{b^2 e 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 )}{3 a^2 d}-\frac{b e^2 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 )}{6 a d^2}-\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )+\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) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{6 a^3}+\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )-\log \left (-\frac{e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{6 a^3}+\frac{b^3 h \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{3 a^3}-\frac{b^2 c \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c) x}+\frac{b^3 c^2 \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}-\frac{b^3 c \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{b^3 c^2 \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}+\frac{b^3 c \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b^3 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 )}{6 a^3}-\frac{e^3 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 )}{6 d^3}+\frac{e^3 h n \log (x) \log \left (1+\frac{b x}{a}\right ) \log (1-c (a+b x))}{3 d^3}+\frac{b^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 a^3}-\frac{e^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 d^3}-\frac{b^3 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}{6 a^3}+\frac{e^3 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}{6 d^3}+\frac{e^3 h n \left (\log \left (1+\frac{b x}{a}\right )+\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )}{6 d^3}+\frac{e^3 h n \left (\log (c (a+b x))-\log \left (1+\frac{b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )^2}{6 d^3}+\frac{e^3 h n \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{3 d^3}-\frac{b^3 g \text{Li}_2(c (a+b x))}{3 a^3}+\frac{b^2 e h n \text{Li}_2(c (a+b x))}{6 a^2 d}+\frac{b e^2 h n \text{Li}_2(c (a+b x))}{3 a d^2}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 a^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{b^3 g \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}+\frac{b^2 e h n \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{2 a^2 d}+\frac{b e^2 h n \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{2 a d^2}-\frac{b^3 h n \left (\log (d+e x)-\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (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 )}{3 a^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}+\frac{b^3 c^2 h n \text{Li}_2\left (1+\frac{e x}{d}\right )}{6 a (1-a c)^2}-\frac{b^3 c h n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^2 (1-a c)}-\frac{b^3 h n \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}+\frac{b^3 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))}{3 a^3}-\frac{e^3 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))}{3 d^3}+\frac{e^3 h n \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2(1-c (a+b x))}{3 d^3}-\frac{b^3 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 )}{3 a^3}+\frac{e^3 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 )}{3 d^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}-\frac{b^3 h n \text{Li}_3(1-c (a+b x))}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 a^3}-\frac{e^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 d^3}+\frac{\left (b^3 c^2 h n\right ) \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 )}{6 a (1-a c)^2}-\frac{\left (b^3 c h n\right ) \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 )}{3 a^2 (1-a c)}+\frac{\left (b^2 e h n\right ) \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 )}{3 a^2 d}+\frac{\left (b^2 c e h n\right ) \int \frac{1}{x} \, dx}{6 a (1-a c) d}-2 \left (-\frac{b e h n \log (1-a c-b c x)}{6 a d x}-\frac{\left (b^2 c e h n\right ) \int \frac{1}{x} \, dx}{6 a (1-a c) d}-\frac{\left (b^3 c^2 e h n\right ) \int \frac{1}{1-a c-b c x} \, dx}{6 a (1-a c) d}\right )+\frac{\left (b e^2 h n\right ) \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 )}{6 a d^2}-\frac{\left (b^2 c e^2 h n\right ) \int \frac{1}{d+e x} \, dx}{6 a (1-a c) d}\\ &=\frac{b^2 c e h n \log (x)}{6 a (1-a c) d}-\frac{b^3 g \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}+\frac{b^2 e h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2 d}+\frac{b e^2 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a d^2}-2 \left (-\frac{b^2 c e h n \log (x)}{6 a (1-a c) d}+\frac{b^2 c e h n \log (1-a c-b c x)}{6 a (1-a c) d}-\frac{b e h n \log (1-a c-b c x)}{6 a d x}\right )-\frac{b^2 c e h n \log (d+e x)}{6 a (1-a c) d}-\frac{b^3 h n \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{3 a^3}-\frac{b^2 e 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 )}{3 a^2 d}-\frac{b e^2 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 )}{6 a d^2}-\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )+\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) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{6 a^3}+\frac{b^3 h n \left (\log \left (\frac{b c x}{1-a c}\right )-\log \left (-\frac{e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{6 a^3}+\frac{b^3 h \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{3 a^3}-\frac{b^2 c \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c) x}+\frac{b^3 c^2 \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}-\frac{b^3 c \log \left (-\frac{e x}{d}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac{b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac{b^3 c^2 \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a (1-a c)^2}+\frac{b^3 c \log \left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 (1-a c)}+\frac{b^3 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 )}{6 a^3}-\frac{e^3 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 )}{6 d^3}+\frac{e^3 h n \log (x) \log \left (1+\frac{b x}{a}\right ) \log (1-c (a+b x))}{3 d^3}+\frac{b^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 a^3}-\frac{e^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 d^3}-\frac{b^3 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}{6 a^3}+\frac{e^3 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}{6 d^3}+\frac{e^3 h n \left (\log \left (1+\frac{b x}{a}\right )+\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )}{6 d^3}+\frac{e^3 h n \left (\log (c (a+b x))-\log \left (1+\frac{b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )^2}{6 d^3}+\frac{e^3 h n \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{3 d^3}-\frac{b^3 g \text{Li}_2(c (a+b x))}{3 a^3}+\frac{b^2 e h n \text{Li}_2(c (a+b x))}{6 a^2 d}+\frac{b e^2 h n \text{Li}_2(c (a+b x))}{3 a d^2}-\frac{e h n \text{Li}_2(c (a+b x))}{6 d x^2}+\frac{e^2 h n \text{Li}_2(c (a+b x))}{3 d^2 x}+\frac{e^3 h n \log (x) \text{Li}_2(c (a+b x))}{3 d^3}-\frac{e^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 d^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 a^3}-\frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))}{3 x^3}-\frac{b^2 e h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 a^2 d}-\frac{b e^2 h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{6 a d^2}-\frac{b^3 g \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}+\frac{b^2 e h n \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{2 a^2 d}+\frac{b e^2 h n \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{2 a d^2}-\frac{b^3 h n \left (\log (d+e x)-\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}+\frac{b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text{Li}_2\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \log \left (\frac{(1-a c) (d+e x)}{d (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 )}{3 a^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{b^3 c^2 h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{6 a (1-a c)^2}+\frac{b^3 c h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{3 a^2 (1-a c)}+\frac{b^3 c^2 h n \text{Li}_2\left (1+\frac{e x}{d}\right )}{6 a (1-a c)^2}-\frac{b^3 c h n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^2 (1-a c)}-\frac{b^3 h n \left (\log (1-a c-b c x)+\log \left (\frac{(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}+\frac{b^3 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))}{3 a^3}-\frac{e^3 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))}{3 d^3}+\frac{e^3 h n \left (\log (x)+\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right ) \text{Li}_2(1-c (a+b x))}{3 d^3}-\frac{b^3 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 )}{3 a^3}+\frac{e^3 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 )}{3 d^3}+\frac{b^3 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 )}{3 a^3}-\frac{e^3 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 )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{3 a^3}+\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-a c-b c x)}{b c (d+e x)}\right )}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{b x}{a (1-c (a+b x))}\right )}{3 d^3}-\frac{e^3 h n \text{Li}_3\left (-\frac{b c x}{1-c (a+b x)}\right )}{3 d^3}-\frac{b^3 h n \text{Li}_3(1-c (a+b x))}{3 a^3}-\frac{b^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 a^3}+\frac{e^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 d^3}+\frac{b^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 a^3}-\frac{e^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 d^3}\\ \end{align*}
Mathematica [F] time = 10.5193, size = 0, normalized size = 0. \[ \int \frac{\left (g+h \log \left (f (d+e x)^n\right )\right ) \text{PolyLog}(2,c (a+b x))}{x^4} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.537, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( g+h\ln \left ( f \left ( ex+d \right ) ^{n} \right ) \right ){\it polylog} \left ( 2,c \left ( bx+a \right ) \right ) }{{x}^{4}}}\, 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*} \int \frac{{\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 )}{x^{4}}\,{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{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^{4}}, 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 (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )}{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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