Optimal. Leaf size=297 \[ \frac {2 B n \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g}-\frac {2 B n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g} \]
[Out]
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Rubi [B] time = 5.18, antiderivative size = 2233, normalized size of antiderivative = 7.52, number of steps used = 43, number of rules used = 21, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.656, Rules used = {2524, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2437, 2435, 2315} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2301
Rule 2315
Rule 2317
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2433
Rule 2435
Rule 2437
Rule 2440
Rule 2500
Rule 2524
Rule 2528
Rule 6589
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{f+g x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{a+b x} \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B (b c-a d) n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B (b c-a d) n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b c-a d) (c+d x)}\right ) \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 b B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {(2 B d n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{c+d x} \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 b B n) \int \left (\frac {A \log (f+g x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)}{a+b x}\right ) \, dx}{g}+\frac {(2 B d n) \int \left (\frac {A \log (f+g x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)}{c+d x}\right ) \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}-\frac {(2 A b B n) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {(2 A B d n) \int \frac {\log (f+g x)}{c+d x} \, dx}{g}+\frac {\left (2 B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+(2 A B n) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-(2 A B n) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (f+g x)}{a+b x} \, dx}{g}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {\left (2 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (f+g x)}{c+d x} \, dx}{g}+\frac {\left (2 B^2 d n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (f+g x)}{c+d x} \, dx}{g}-\frac {\left (2 b B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}+\frac {\left (2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}+\frac {(2 A B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {(2 A B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b f-a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right ) \log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {d f-c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {b c-a d}{d}+\frac {b x}{d}\right )^n\right ) \log \left (-\frac {-d f+c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}+\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log (f+g x)}{g}+\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 A B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b f-a g}{b}+\frac {g x}{b}} \, dx,x,a+b x\right )}{b}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {d f-c g}{d}+\frac {g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d}+\frac {b x}{d}\right ) \log \left (-\frac {-d f+c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}-\frac {\left (2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-d f+c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}+\frac {\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {\left (2 B^2 n \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}+\frac {2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log (f+g x)}{g}+\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {g x}{b f-a g}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {g x}{d f-c g}\right )}{x} \, dx,x,c+d x\right )}{g}+\frac {\left (2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {g x}{-d f+c g}\right )}{-\frac {-d f+c g}{d}+\frac {g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (2 B^2 n \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {g x}{-b f+a g}\right )}{-\frac {-b f+a g}{b}+\frac {g x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}+\frac {2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log (f+g x)}{g}+\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 n \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{b f-a g}\right )}{x} \, dx,x,a+b x\right )}{g}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{d f-c g}\right )}{x} \, dx,x,c+d x\right )}{g}\\ &=-\frac {2 A B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}+\frac {2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log (f+g x)}{g}+\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}-\frac {2 B^2 n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 n^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 n \left (n \log (c+d x)+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n \left (n \log (a+b x)-\log \left ((a+b x)^n\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 n^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}\\ \end {align*}
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Mathematica [B] time = 0.46, size = 1441, normalized size = 4.85 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{g x +f}\, dx \]
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 \[ \frac {A^{2} \log \left (g x + f\right )}{g} + \int \frac {B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + B^{2} \log \relax (e)^{2} + 2 \, A B \log \relax (e) + 2 \, {\left (B^{2} \log \relax (e) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \relax (e) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{g x + f}\,{d x} \]
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 \[ \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\right )^{2}}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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