Optimal. Leaf size=2252 \[ \text {result too large to display} \]
[Out]
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Rubi [A] time = 2.90, antiderivative size = 2252, normalized size of antiderivative = 1.00, number of steps used = 67, number of rules used = 20, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 2421
Rule 2430
Rule 2435
Rule 2437
Rule 2438
Rule 2439
Rule 2440
Rule 2444
Rule 6595
Rule 6597
Rule 6598
Rule 6603
Rubi steps
\begin {align*} \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x)) \, dx &=\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {1}{2} b \int \left (-\frac {a \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2}+\frac {x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b}+\frac {a^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 (a+b x)}\right ) \, dx-\frac {1}{2} (e h n) \int \left (-\frac {d \text {Li}_2(c (a+b x))}{e^2}+\frac {x \text {Li}_2(c (a+b x))}{e}+\frac {d^2 \text {Li}_2(c (a+b x))}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {1}{2} \int x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx-\frac {a \int \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{2 b}+\frac {a^2 \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}{2 b}-\frac {1}{2} (h n) \int x \text {Li}_2(c (a+b x)) \, dx+\frac {(d h n) \int \text {Li}_2(c (a+b x)) \, dx}{2 e}-\frac {\left (d^2 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx}{2 e}\\ &=-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {a^2 \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 )}{2 b^2}-\frac {1}{2} (a c) \int \frac {x \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx+\frac {1}{4} (b 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 {1}{4} (b h n) \int \frac {x^2 \log (1-a c-b c x)}{a+b x} \, dx-\frac {\left (b d^2 h n\right ) \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{2 e^2}+\frac {(d h n) \int \log (1-c (a+b x)) \, dx}{2 e}-\frac {(a d h n) \int \frac {\log (1-c (a+b x))}{a+b x} \, dx}{2 e}-\frac {1}{4} (e h n) \int \frac {x^2 \log (1-a c-b c x)}{d+e x} \, dx+\frac {(a e h n) \int \frac {x \log (1-a c-b c x)}{d+e x} \, dx}{2 b}\\ &=-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {1}{2} (a c) \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+\frac {1}{4} (b 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 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 )}{2 b^2}+\frac {\left (a^2 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 )}{2 b^2}-\frac {1}{4} (b 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-\frac {\left (d^2 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 )}{2 e^2}+\frac {(d h n) \int \log (1-a c-b c x) \, dx}{2 e}-\frac {(a d h n) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{2 e}-\frac {1}{4} (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+\frac {(a e h n) \int \left (\frac {\log (1-a c-b c x)}{e}-\frac {d \log (1-a c-b c x)}{e (d+e x)}\right ) \, dx}{2 b}\\ &=-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {d^2 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 )}{4 e^2}-\frac {d^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 e^2}+\frac {d^2 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}{4 e^2}-\frac {a^2 g \text {Li}_2(c (a+b x))}{2 b^2}+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {d^2 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 )}{2 e^2}-\frac {d^2 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))}{2 e^2}+\frac {d^2 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 )}{2 e^2}-\frac {d^2 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 )}{2 e^2}+\frac {d^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 e^2}+\frac {d^2 h n \text {Li}_3(1-c (a+b x))}{2 e^2}+\frac {d^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 e^2}-\frac {d^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 e^2}-\frac {1}{4} \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx+\frac {a \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{2 b}+\frac {(a (1-a c)) \int \frac {g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{2 b}-\frac {(1-a c) \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{4 b c}-\frac {(1-a c)^2 \int \frac {g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{4 b c}-2 \left (\frac {1}{4} (h n) \int x \log (1-a c-b c x) \, dx\right )+\frac {\left (a^2 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 )}{2 b^2}+\frac {(a h n) \int \log (1-a c-b c x) \, dx}{4 b}+\frac {(a h n) \int \log (1-a c-b c x) \, dx}{2 b}-\frac {\left (a^2 h n\right ) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{4 b}-\frac {(a d h n) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{2 b}+\frac {(d h n) \int \log (1-a c-b c x) \, dx}{4 e}-\frac {(a d h n) \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 b e}-\frac {(d h n) \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{2 b c e}-\frac {\left (d^2 h n\right ) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{4 e}-\frac {\left (a^2 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 )}{2 b^2}\\ &=\frac {a g x}{2 b}-\frac {(1-a c) g x}{4 b c}-\frac {d h n x}{2 e}-\frac {d h n (1-a c-b c x) \log (1-a c-b c x)}{2 b c e}-\frac {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 )}{4 e^2}-\frac {a 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 )}{2 b e}-\frac {1}{8} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a (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 )}{2 b^2 c}-\frac {(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 )}{4 b^2 c^2}+\frac {a^2 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 )}{4 b^2}-\frac {d^2 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 )}{4 e^2}+\frac {a^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 b^2}-\frac {d^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 e^2}-\frac {a^2 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}{4 b^2}+\frac {d^2 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}{4 e^2}-\frac {a^2 g \text {Li}_2(c (a+b x))}{2 b^2}+\frac {a d h n \text {Li}_2(c (a+b x))}{2 b e}+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {a^2 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{2 b^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}+\frac {a^2 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))}{2 b^2}-\frac {d^2 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))}{2 e^2}-\frac {a^2 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 )}{2 b^2}+\frac {d^2 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 )}{2 e^2}+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 e^2}-\frac {a^2 h n \text {Li}_3(1-c (a+b x))}{2 b^2}+\frac {d^2 h n \text {Li}_3(1-c (a+b x))}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 e^2}+\frac {a^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 b^2}-\frac {d^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 e^2}+\frac {(a h) \int \log \left (f (d+e x)^n\right ) \, dx}{2 b}-\frac {((1-a c) h) \int \log \left (f (d+e x)^n\right ) \, dx}{4 b c}-\frac {\left (a^2 h n\right ) \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{4 b^2}-\frac {(a h n) \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{4 b^2 c}-\frac {(a h n) \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{2 b^2 c}-2 \left (\frac {1}{8} h n x^2 \log (1-a c-b c x)+\frac {1}{8} (b c h n) \int \frac {x^2}{1-a c-b c x} \, dx\right )-\frac {\left (b 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}{4 e^2}-\frac {(d h n) \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{4 b c e}-\frac {(a c d h n) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{2 e}+\frac {1}{8} (e h n) \int \frac {x^2}{d+e x} \, dx-\frac {(a (1-a c) e h n) \int \frac {\log \left (\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{2 b^2 c}+\frac {\left ((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}{4 b^2 c^2}\\ &=\frac {a g x}{2 b}-\frac {(1-a c) g x}{4 b c}-\frac {3 a h n x}{4 b}-\frac {3 d h n x}{4 e}-\frac {3 a h n (1-a c-b c x) \log (1-a c-b c x)}{4 b^2 c}-\frac {3 d h n (1-a c-b c x) \log (1-a c-b c x)}{4 b c e}-\frac {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 )}{4 e^2}-\frac {a 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 )}{2 b e}-\frac {1}{8} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a (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 )}{2 b^2 c}-\frac {(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 )}{4 b^2 c^2}+\frac {a^2 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 )}{4 b^2}-\frac {d^2 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 )}{4 e^2}+\frac {a^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 b^2}-\frac {d^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 e^2}-\frac {a^2 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}{4 b^2}+\frac {d^2 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}{4 e^2}-\frac {a^2 g \text {Li}_2(c (a+b x))}{2 b^2}+\frac {a^2 h n \text {Li}_2(c (a+b x))}{4 b^2}+\frac {a d h n \text {Li}_2(c (a+b x))}{2 b e}+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {a^2 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{2 b^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}+\frac {a^2 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))}{2 b^2}-\frac {d^2 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))}{2 e^2}-\frac {a^2 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 )}{2 b^2}+\frac {d^2 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 )}{2 e^2}+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 e^2}-\frac {a^2 h n \text {Li}_3(1-c (a+b x))}{2 b^2}+\frac {d^2 h n \text {Li}_3(1-c (a+b x))}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 e^2}+\frac {a^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 b^2}-\frac {d^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 e^2}+\frac {(a h) \operatorname {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{2 b e}-\frac {((1-a c) h) \operatorname {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{4 b c e}-2 \left (\frac {1}{8} h n x^2 \log (1-a c-b c x)+\frac {1}{8} (b 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\right )-\frac {(a (1-a c) h n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{2 b^2 c}+\frac {\left ((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 )}{4 b^2 c^2}+\frac {\left (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 )}{4 e^2}+\frac {(a d h n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{2 b e}+\frac {1}{8} (e h n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {a g x}{2 b}-\frac {(1-a c) g x}{4 b c}-\frac {5 a h n x}{4 b}+\frac {(1-a c) h n x}{4 b c}-\frac {7 d h n x}{8 e}+\frac {1}{16} h n x^2-\frac {3 a h n (1-a c-b c x) \log (1-a c-b c x)}{4 b^2 c}-\frac {3 d h n (1-a c-b c x) \log (1-a c-b c x)}{4 b c e}-2 \left (-\frac {(1-a c) h n x}{8 b c}-\frac {1}{16} h n x^2-\frac {(1-a c)^2 h n \log (1-a c-b c x)}{8 b^2 c^2}+\frac {1}{8} h n x^2 \log (1-a c-b c x)\right )+\frac {d^2 h n \log (d+e x)}{8 e^2}-\frac {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 )}{4 e^2}-\frac {a 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 )}{2 b e}+\frac {a h (d+e x) \log \left (f (d+e x)^n\right )}{2 b e}-\frac {(1-a c) h (d+e x) \log \left (f (d+e x)^n\right )}{4 b c e}-\frac {1}{8} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}+\frac {1}{4} x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a (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 )}{2 b^2 c}-\frac {(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 )}{4 b^2 c^2}+\frac {a^2 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 )}{4 b^2}-\frac {d^2 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 )}{4 e^2}+\frac {a^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 b^2}-\frac {d^2 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{2 e^2}-\frac {a^2 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}{4 b^2}+\frac {d^2 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}{4 e^2}-\frac {a^2 g \text {Li}_2(c (a+b x))}{2 b^2}+\frac {a^2 h n \text {Li}_2(c (a+b x))}{4 b^2}+\frac {a d h n \text {Li}_2(c (a+b x))}{2 b e}+\frac {d h n x \text {Li}_2(c (a+b x))}{2 e}-\frac {1}{4} h n x^2 \text {Li}_2(c (a+b x))-\frac {d^2 h n \log (d+e x) \text {Li}_2(c (a+b x))}{2 e^2}+\frac {a^2 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{2 b^2}+\frac {1}{2} x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {d^2 h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{4 e^2}-\frac {a d h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{2 b e}+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}+\frac {a (1-a c) h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 b^2 c}-\frac {(1-a c)^2 h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{4 b^2 c^2}+\frac {a^2 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))}{2 b^2}-\frac {d^2 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))}{2 e^2}-\frac {a^2 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 )}{2 b^2}+\frac {d^2 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 )}{2 e^2}+\frac {a^2 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 )}{2 b^2}-\frac {d^2 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 )}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{2 e^2}-\frac {a^2 h n \text {Li}_3(1-c (a+b x))}{2 b^2}+\frac {d^2 h n \text {Li}_3(1-c (a+b x))}{2 e^2}-\frac {a^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 b^2}+\frac {d^2 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{2 e^2}+\frac {a^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 b^2}-\frac {d^2 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{2 e^2}\\ \end {align*}
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Mathematica [A] time = 6.16, size = 1996, normalized size = 0.89 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (h x {\rm Li}_2\left (b c x + a c\right ) \log \left ({\left (e x + d\right )}^{n} f\right ) + g x {\rm Li}_2\left (b c x + a c\right ), x\right ) \]
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 \[ \int {\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} x {\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x \left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )\, dx \]
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 \[ \frac {{\left (2 \, e^{2} h x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + 2 \, d e h n x - 2 \, d^{2} h n \log \left (e x + d\right ) - {\left (e^{2} h n - 2 \, e^{2} h \log \relax (f) - 2 \, e^{2} g\right )} x^{2}\right )} {\rm Li}_2\left (b c x + a c\right )}{4 \, e^{2}} + \int \frac {2 \, b e^{2} h x^{2} \log \left (-b c x - a c + 1\right ) \log \left ({\left (e x + d\right )}^{n}\right ) + {\left (2 \, b d e h n x - 2 \, b d^{2} h n \log \left (e x + d\right ) - {\left (b e^{2} h n - 2 \, b e^{2} h \log \relax (f) - 2 \, b e^{2} g\right )} x^{2}\right )} \log \left (-b c x - a c + 1\right )}{4 \, {\left (b e^{2} x + a e^{2}\right )}}\,{d x} \]
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 \[ \int x\,\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (g+h\,\ln \left (f\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \]
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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