Optimal. Leaf size=3733 \[ \text {result too large to display} \]
[Out]
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Rubi [A] time = 4.31, antiderivative size = 3733, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 44
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 2435
Rule 2437
Rule 2438
Rule 2439
Rule 2440
Rule 6597
Rule 6598
Rule 6603
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*}
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Mathematica [F] time = 10.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x^4} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {h {\rm Li}_2\left (b c x + a c\right ) \log \left ({\left (e x + d\right )}^{n} f\right ) + g {\rm Li}_2\left (b c x + a c\right )}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (g+h\,\ln \left (f\,{\left (d+e\,x\right )}^n\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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