Optimal. Leaf size=2995 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 4.59961, antiderivative size = 2995, normalized size of antiderivative = 1., number of steps used = 108, number of rules used = 20, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.741, Rules used = {6603, 2430, 43, 2416, 2389, 2295, 2394, 2393, 2391, 2439, 2395, 2440, 2438, 2437, 2435, 6595, 2444, 2421, 6598, 6597} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6603
Rule 2430
Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2439
Rule 2395
Rule 2440
Rule 2438
Rule 2437
Rule 2435
Rule 6595
Rule 2444
Rule 2421
Rule 6598
Rule 6597
Rubi steps
\begin{align*} \int x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x)) \, dx &=\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{1}{3} b \int \left (\frac{a^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3}-\frac{a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2}+\frac{x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b}-\frac{a^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 (a+b x)}\right ) \, dx-\frac{1}{3} (e h n) \int \left (\frac{d^2 \text{Li}_2(c (a+b x))}{e^3}-\frac{d x \text{Li}_2(c (a+b x))}{e^2}+\frac{x^2 \text{Li}_2(c (a+b x))}{e}-\frac{d^3 \text{Li}_2(c (a+b x))}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{1}{3} \int x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx+\frac{a^2 \int \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b^2}-\frac{a^3 \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 b^2}-\frac{a \int x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b}-\frac{1}{3} (h n) \int x^2 \text{Li}_2(c (a+b x)) \, dx-\frac{\left (d^2 h n\right ) \int \text{Li}_2(c (a+b x)) \, dx}{3 e^2}+\frac{\left (d^3 h n\right ) \int \frac{\text{Li}_2(c (a+b x))}{d+e x} \, dx}{3 e^2}+\frac{(d h n) \int x \text{Li}_2(c (a+b x)) \, dx}{3 e}\\ &=\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-\frac{a^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 b^3}-\frac{1}{6} (a c) \int \frac{x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx+\frac{\left (a^2 c\right ) \int \frac{x \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx}{3 b}+\frac{1}{9} (b c) \int \frac{x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx-\frac{1}{9} (b h n) \int \frac{x^3 \log (1-a c-b c x)}{a+b x} \, dx+\frac{\left (b d^3 h n\right ) \int \frac{\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{3 e^3}-\frac{\left (d^2 h n\right ) \int \log (1-c (a+b x)) \, dx}{3 e^2}+\frac{\left (a d^2 h n\right ) \int \frac{\log (1-c (a+b x))}{a+b x} \, dx}{3 e^2}+\frac{(b d h n) \int \frac{x^2 \log (1-a c-b c x)}{a+b x} \, dx}{6 e}-\frac{1}{9} (e h n) \int \frac{x^3 \log (1-a c-b c x)}{d+e x} \, dx-\frac{\left (a^2 e h n\right ) \int \frac{x \log (1-a c-b c x)}{d+e x} \, dx}{3 b^2}+\frac{(a e h n) \int \frac{x^2 \log (1-a c-b c x)}{d+e x} \, dx}{6 b}\\ &=\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-\frac{1}{6} (a c) \int \left (\frac{(-1+a c) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2}-\frac{x \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c}-\frac{(-1+a c)^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx+\frac{\left (a^2 c\right ) \int \left (-\frac{g+h \log \left (f (d+e x)^n\right )}{b c}+\frac{(-1+a c) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c (-1+a c+b c x)}\right ) \, dx}{3 b}+\frac{1}{9} (b c) \int \left (-\frac{(-1+a c)^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 c^3}+\frac{(-1+a c) x \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2}-\frac{x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c}+\frac{(-1+a c)^3 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 c^3 (-1+a c+b c x)}\right ) \, dx-\frac{\left (a^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 b^3}-\frac{\left (a^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 b^3}-\frac{1}{9} (b h n) \int \left (\frac{a^2 \log (1-a c-b c x)}{b^3}-\frac{a x \log (1-a c-b c x)}{b^2}+\frac{x^2 \log (1-a c-b c x)}{b}-\frac{a^3 \log (1-a c-b c x)}{b^3 (a+b x)}\right ) \, dx+\frac{\left (d^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 e^3}-\frac{\left (d^2 h n\right ) \int \log (1-a c-b c x) \, dx}{3 e^2}+\frac{\left (a d^2 h n\right ) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{3 e^2}+\frac{(b d h n) \int \left (-\frac{a \log (1-a c-b c x)}{b^2}+\frac{x \log (1-a c-b c x)}{b}+\frac{a^2 \log (1-a c-b c x)}{b^2 (a+b x)}\right ) \, dx}{6 e}-\frac{1}{9} (e h n) \int \left (\frac{d^2 \log (1-a c-b c x)}{e^3}-\frac{d x \log (1-a c-b c x)}{e^2}+\frac{x^2 \log (1-a c-b c x)}{e}-\frac{d^3 \log (1-a c-b c x)}{e^3 (d+e x)}\right ) \, dx-\frac{\left (a^2 e h n\right ) \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}{3 b^2}+\frac{(a e h n) \int \left (-\frac{d \log (1-a c-b c x)}{e^2}+\frac{x \log (1-a c-b c x)}{e}+\frac{d^2 \log (1-a c-b c x)}{e^2 (d+e x)}\right ) \, dx}{6 b}\\ &=\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac{d^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 e^3}+\frac{d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}-\frac{d^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 e^3}+\frac{a^3 g \text{Li}_2(c (a+b x))}{3 b^3}-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d^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 e^3}+\frac{d^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 e^3}-\frac{d^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 e^3}+\frac{d^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 e^3}-\frac{d^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 e^3}-\frac{d^3 h n \text{Li}_3(1-c (a+b x))}{3 e^3}-\frac{d^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}+\frac{d^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac{1}{9} \int x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx-\frac{a^2 \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b^2}+\frac{a \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{6 b}-\frac{\left (a^2 (1-a c)\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{3 b^2}+\frac{(a (1-a c)) \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{6 b^2 c}-\frac{(1-a c) \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{9 b c}-\frac{(1-a c)^2 \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{9 b^2 c^2}+\frac{\left (a (1-a c)^2\right ) \int \frac{g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{6 b^2 c}-\frac{(1-a c)^3 \int \frac{g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{9 b^2 c^2}-2 \left (\frac{1}{9} (h n) \int x^2 \log (1-a c-b c x) \, dx\right )-\frac{\left (a^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 b^3}-\frac{\left (a^2 h n\right ) \int \log (1-a c-b c x) \, dx}{9 b^2}-\frac{\left (a^2 h n\right ) \int \log (1-a c-b c x) \, dx}{3 b^2}+\frac{\left (a^3 h n\right ) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{9 b^2}+\frac{(a h n) \int x \log (1-a c-b c x) \, dx}{9 b}+\frac{(a h n) \int x \log (1-a c-b c x) \, dx}{6 b}+\frac{\left (a^2 d h n\right ) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{3 b^2}-\frac{\left (d^2 h n\right ) \int \log (1-a c-b c x) \, dx}{9 e^2}+\frac{\left (a d^2 h n\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 b e^2}+\frac{\left (d^2 h n\right ) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b c e^2}+\frac{\left (d^3 h n\right ) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{9 e^2}+\frac{(d h n) \int x \log (1-a c-b c x) \, dx}{9 e}+\frac{(d h n) \int x \log (1-a c-b c x) \, dx}{6 e}-2 \frac{(a d h n) \int \log (1-a c-b c x) \, dx}{6 b e}+\frac{\left (a^2 d h n\right ) \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{6 b e}+\frac{\left (a d^2 h n\right ) \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{6 b e}+\frac{\left (a^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 b^3}\\ &=-\frac{a^2 g x}{3 b^2}+\frac{a (1-a c) g x}{6 b^2 c}-\frac{(1-a c)^2 g x}{9 b^2 c^2}+\frac{d^2 h n x}{3 e^2}+\frac{5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac{5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac{d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{3 b c e^2}+\frac{d^3 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 )}{9 e^3}+\frac{a d^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 b e^2}+\frac{a^2 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 )}{3 b^2 e}+\frac{a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac{(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac{1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac{a^2 (1-a 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 b^3 c}+\frac{a (1-a 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 b^3 c^2}-\frac{(1-a c)^3 \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 )}{9 b^3 c^3}-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac{d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}+\frac{a^3 g \text{Li}_2(c (a+b x))}{3 b^3}-\frac{a d^2 h n \text{Li}_2(c (a+b x))}{3 b e^2}-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}-\frac{a^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 b^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac{a^3 h n \text{Li}_3(1-c (a+b x))}{3 b^3}-\frac{d^3 h n \text{Li}_3(1-c (a+b x))}{3 e^3}+\frac{a^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac{a^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac{d^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac{\left (a^2 h\right ) \int \log \left (f (d+e x)^n\right ) \, dx}{3 b^2}+\frac{(a (1-a c) h) \int \log \left (f (d+e x)^n\right ) \, dx}{6 b^2 c}-\frac{\left ((1-a c)^2 h\right ) \int \log \left (f (d+e x)^n\right ) \, dx}{9 b^2 c^2}+\frac{\left (a^3 h n\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{9 b^3}+\frac{\left (a^2 h n\right ) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{9 b^3 c}+\frac{\left (a^2 h n\right ) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}+\frac{1}{18} (a c h n) \int \frac{x^2}{1-a c-b c x} \, dx+\frac{1}{12} (a c h n) \int \frac{x^2}{1-a c-b c x} \, dx-2 \left (\frac{1}{27} h n x^3 \log (1-a c-b c x)+\frac{1}{27} (b c h n) \int \frac{x^3}{1-a c-b c x} \, dx\right )+\frac{\left (b c d^3 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}{9 e^3}+\frac{\left (d^2 h n\right ) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{9 b c e^2}+\frac{\left (a c d^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 e^2}+\frac{\left (a^2 d h n\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{6 b^2 e}+2 \frac{(a d h n) \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{6 b^2 c e}+\frac{\left (a^2 c d 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 b e}+\frac{(b c d h n) \int \frac{x^2}{1-a c-b c x} \, dx}{18 e}+\frac{(b c d h n) \int \frac{x^2}{1-a c-b c x} \, dx}{12 e}+\frac{1}{27} (e h n) \int \frac{x^3}{d+e x} \, dx-\frac{(a e h n) \int \frac{x^2}{d+e x} \, dx}{12 b}+\frac{\left (a^2 (1-a 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 b^3 c}+\frac{((1-a c) e h n) \int \frac{x^2}{d+e x} \, dx}{18 b c}-\frac{\left (a (1-a 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 b^3 c^2}+\frac{\left ((1-a c)^3 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}{9 b^3 c^3}\\ &=-\frac{a^2 g x}{3 b^2}+\frac{a (1-a c) g x}{6 b^2 c}-\frac{(1-a c)^2 g x}{9 b^2 c^2}+\frac{4 a^2 h n x}{9 b^2}+\frac{4 d^2 h n x}{9 e^2}+\frac{5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac{5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac{4 a^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b^3 c}+\frac{4 d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b c e^2}+2 \left (\frac{a d h n x}{6 b e}+\frac{a d h n (1-a c-b c x) \log (1-a c-b c x)}{6 b^2 c e}\right )+\frac{d^3 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 )}{9 e^3}+\frac{a d^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 b e^2}+\frac{a^2 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 )}{3 b^2 e}+\frac{a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac{(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac{1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac{a^2 (1-a 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 b^3 c}+\frac{a (1-a 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 b^3 c^2}-\frac{(1-a c)^3 \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 )}{9 b^3 c^3}-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac{d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}+\frac{a^3 g \text{Li}_2(c (a+b x))}{3 b^3}-\frac{a^3 h n \text{Li}_2(c (a+b x))}{9 b^3}-\frac{a d^2 h n \text{Li}_2(c (a+b x))}{3 b e^2}-\frac{a^2 d h n \text{Li}_2(c (a+b x))}{6 b^2 e}-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}-\frac{a^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 b^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac{a^3 h n \text{Li}_3(1-c (a+b x))}{3 b^3}-\frac{d^3 h n \text{Li}_3(1-c (a+b x))}{3 e^3}+\frac{a^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac{a^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac{d^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac{\left (a^2 h\right ) \operatorname{Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{3 b^2 e}+\frac{(a (1-a c) h) \operatorname{Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{6 b^2 c e}-\frac{\left ((1-a c)^2 h\right ) \operatorname{Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{9 b^2 c^2 e}+\frac{1}{18} (a c h n) \int \left (\frac{-1+a c}{b^2 c^2}-\frac{x}{b c}-\frac{(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx+\frac{1}{12} (a c h n) \int \left (\frac{-1+a c}{b^2 c^2}-\frac{x}{b c}-\frac{(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx-2 \left (\frac{1}{27} h n x^3 \log (1-a c-b c x)+\frac{1}{27} (b c h n) \int \left (-\frac{(-1+a c)^2}{b^3 c^3}+\frac{(-1+a c) x}{b^2 c^2}-\frac{x^2}{b c}+\frac{(-1+a c)^3}{b^3 c^3 (-1+a c+b c x)}\right ) \, dx\right )+\frac{\left (a^2 (1-a 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 b^3 c}-\frac{\left (a (1-a 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 b^3 c^2}+\frac{\left ((1-a c)^3 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 )}{9 b^3 c^3}-\frac{\left (d^3 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 )}{9 e^3}-\frac{\left (a d^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 b e^2}-\frac{\left (a^2 d 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 b^2 e}+\frac{(b c d h n) \int \left (\frac{-1+a c}{b^2 c^2}-\frac{x}{b c}-\frac{(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx}{18 e}+\frac{(b c d h n) \int \left (\frac{-1+a c}{b^2 c^2}-\frac{x}{b c}-\frac{(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx}{12 e}+\frac{1}{27} (e h n) \int \left (\frac{d^2}{e^3}-\frac{d x}{e^2}+\frac{x^2}{e}-\frac{d^3}{e^3 (d+e x)}\right ) \, dx-\frac{(a e h n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx}{12 b}+\frac{((1-a c) e h n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx}{18 b c}\\ &=-\frac{a^2 g x}{3 b^2}+\frac{a (1-a c) g x}{6 b^2 c}-\frac{(1-a c)^2 g x}{9 b^2 c^2}+\frac{7 a^2 h n x}{9 b^2}-\frac{11 a (1-a c) h n x}{36 b^2 c}+\frac{(1-a c)^2 h n x}{9 b^2 c^2}+\frac{13 d^2 h n x}{27 e^2}+\frac{a d h n x}{12 b e}-\frac{7 (1-a c) d h n x}{36 b c e}-\frac{a h n x^2}{9 b}+\frac{(1-a c) h n x^2}{36 b c}-\frac{19 d h n x^2}{216 e}+\frac{1}{81} h n x^3-\frac{5 a (1-a c)^2 h n \log (1-a c-b c x)}{36 b^3 c^2}-\frac{5 (1-a c)^2 d h n \log (1-a c-b c x)}{36 b^2 c^2 e}+\frac{5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac{5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac{4 a^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b^3 c}+\frac{4 d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b c e^2}-2 \left (-\frac{(1-a c)^2 h n x}{27 b^2 c^2}-\frac{(1-a c) h n x^2}{54 b c}-\frac{1}{81} h n x^3-\frac{(1-a c)^3 h n \log (1-a c-b c x)}{27 b^3 c^3}+\frac{1}{27} h n x^3 \log (1-a c-b c x)\right )+2 \left (\frac{a d h n x}{6 b e}+\frac{a d h n (1-a c-b c x) \log (1-a c-b c x)}{6 b^2 c e}\right )-\frac{d^3 h n \log (d+e x)}{27 e^3}-\frac{a d^2 h n \log (d+e x)}{12 b e^2}+\frac{(1-a c) d^2 h n \log (d+e x)}{18 b c e^2}+\frac{d^3 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 )}{9 e^3}+\frac{a d^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 b e^2}+\frac{a^2 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 )}{3 b^2 e}-\frac{a^2 h (d+e x) \log \left (f (d+e x)^n\right )}{3 b^2 e}+\frac{a (1-a c) h (d+e x) \log \left (f (d+e x)^n\right )}{6 b^2 c e}-\frac{(1-a c)^2 h (d+e x) \log \left (f (d+e x)^n\right )}{9 b^2 c^2 e}+\frac{a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac{(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac{1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac{a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac{a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac{1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac{a^2 (1-a 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 b^3 c}+\frac{a (1-a 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 b^3 c^2}-\frac{(1-a c)^3 \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 )}{9 b^3 c^3}-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac{d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}+\frac{a^3 g \text{Li}_2(c (a+b x))}{3 b^3}-\frac{a^3 h n \text{Li}_2(c (a+b x))}{9 b^3}-\frac{a d^2 h n \text{Li}_2(c (a+b x))}{3 b e^2}-\frac{a^2 d h n \text{Li}_2(c (a+b x))}{6 b^2 e}-\frac{d^2 h n x \text{Li}_2(c (a+b x))}{3 e^2}+\frac{d h n x^2 \text{Li}_2(c (a+b x))}{6 e}-\frac{1}{9} h n x^3 \text{Li}_2(c (a+b x))+\frac{d^3 h n \log (d+e x) \text{Li}_2(c (a+b x))}{3 e^3}-\frac{a^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 b^3}+\frac{1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text{Li}_2(c (a+b x))+\frac{d^3 h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{9 e^3}+\frac{a d^2 h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{6 b e^2}+\frac{a^2 d h n \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 b^2 e}-\frac{a^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 b^3}+\frac{d^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 e^3}-\frac{a^2 (1-a c) h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{3 b^3 c}+\frac{a (1-a c)^2 h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{6 b^3 c^2}-\frac{(1-a c)^3 h n \text{Li}_2\left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{9 b^3 c^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^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 b^3}-\frac{d^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 e^3}-\frac{a^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 b^3}+\frac{d^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 e^3}+\frac{a^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (\frac{b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac{a^3 h n \text{Li}_3(1-c (a+b x))}{3 b^3}-\frac{d^3 h n \text{Li}_3(1-c (a+b x))}{3 e^3}+\frac{a^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac{d^3 h n \text{Li}_3\left (-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac{a^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac{d^3 h n \text{Li}_3\left (\frac{(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}\\ \end{align*}
Mathematica [A] time = 11.4813, size = 2610, normalized size = 0.87 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.493, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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 (6 \, e^{3} h x^{3} \log \left ({\left (e x + d\right )}^{n}\right ) + 3 \, d e^{2} h n x^{2} - 6 \, d^{2} e h n x + 6 \, d^{3} h n \log \left (e x + d\right ) - 2 \,{\left (e^{3} h n - 3 \, e^{3} h \log \left (f\right ) - 3 \, e^{3} g\right )} x^{3}\right )}{\rm Li}_2\left (b c x + a c\right )}{18 \, e^{3}} + \int \frac{6 \, b e^{3} h x^{3} \log \left (-b c x - a c + 1\right ) \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (3 \, b d e^{2} h n x^{2} - 6 \, b d^{2} e h n x + 6 \, b d^{3} h n \log \left (e x + d\right ) - 2 \,{\left (b e^{3} h n - 3 \, b e^{3} h \log \left (f\right ) - 3 \, b e^{3} g\right )} x^{3}\right )} \log \left (-b c x - a c + 1\right )}{18 \,{\left (b e^{3} x + a e^{3}\right )}}\,{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 x^{2}{\rm Li}_2\left (b c x + a c\right ) \log \left ({\left (e x + d\right )}^{n} f\right ) + g x^{2}{\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 )} x^{2}{\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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