Optimal. Leaf size=306 \[ \frac {2 B i n (b c-a d) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac {2 B i n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac {d i (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g}-\frac {i (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g}+\frac {2 B^2 i n^2 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 g}+\frac {2 B^2 i n^2 (b c-a d) \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g} \]
[Out]
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Rubi [B] time = 2.87, antiderivative size = 692, normalized size of antiderivative = 2.26, number of steps used = 36, number of rules used = 19, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac {2 A B i n (b c-a d) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {2 B^2 i n (b c-a d) \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {2 a B^2 d i n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {2 B^2 i n^2 (b c-a d) \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b^2 g}+\frac {2 B^2 c i n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 a B d i n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac {i (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g}-\frac {A B i n (b c-a d) \log ^2(a+b x)}{b^2 g}+\frac {2 A B i n (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {2 B c i n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g}+\frac {d i x \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g}-\frac {B^2 i (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac {B^2 i (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {2 a B^2 d i n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {a B^2 d i n^2 \log ^2(a+b x)}{b^2 g}+\frac {2 B^2 c i n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {B^2 c i n^2 \log ^2(c+d x)}{b g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2301
Rule 2317
Rule 2344
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2488
Rule 2506
Rule 2507
Rule 2523
Rule 2524
Rule 2528
Rule 6610
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(163 c+163 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac {163 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g (a+b x)}\right ) \, dx\\ &=\frac {(163 d) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}+\frac {(163 (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{b g}\\ &=\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {(326 B d n) \int \frac {(b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac {(326 B (b c-a d) n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b^2 g}\\ &=\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {(326 B (b c-a d) n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}-\frac {(326 B d (b c-a d) n) \int \frac {x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {(326 B d (b c-a d) n) \int \left (-\frac {a \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b g}-\frac {\left (326 B (b c-a d)^2 n\right ) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {(326 a B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}-\frac {(326 B c d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b g}-\frac {\left (326 B (b c-a d)^2 n\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^2 g}\\ &=\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac {\left (326 A B (b c-a d)^2 n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g}-\frac {\left (326 B^2 (b c-a d)^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}+\frac {\left (326 B^2 c n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b g}-\frac {\left (326 a B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g}\\ &=-\frac {163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}+\frac {\left (163 B^2 (b c-a d)\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}-\frac {\left (326 A B (b c-a d)^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 g}+\frac {\left (326 B^2 c n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b g}-\frac {\left (326 a B^2 d n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g}\\ &=-\frac {163 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac {163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac {(326 A B (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac {(326 A B d (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^3 g}+\frac {\left (326 B^2 (b c-a d)^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}+\frac {\left (326 B^2 c n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{g}-\frac {\left (326 a B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g}-\frac {\left (326 B^2 c d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b g}+\frac {\left (326 a B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g}\\ &=-\frac {163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac {163 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac {163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {326 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}+\frac {326 A B (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 a B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 B^2 (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac {(326 A B (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac {\left (326 B^2 c n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (326 a B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac {\left (326 a B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}-\frac {\left (326 B^2 c d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b g}-\frac {\left (326 B^2 (b c-a d)^2 n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=-\frac {163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac {163 a B^2 d n^2 \log ^2(a+b x)}{b^2 g}-\frac {163 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac {163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {326 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac {163 B^2 c n^2 \log ^2(c+d x)}{b g}+\frac {326 A B (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 a B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 A B (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {326 B^2 (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac {326 B^2 (b c-a d) n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac {\left (326 B^2 c n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (326 a B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}\\ &=-\frac {163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac {163 a B^2 d n^2 \log ^2(a+b x)}{b^2 g}-\frac {163 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac {163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac {326 a B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {163 d x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {163 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {326 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {326 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac {163 B^2 c n^2 \log ^2(c+d x)}{b g}+\frac {326 A B (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 a B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {326 A B (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {326 a B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {326 B^2 c n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {326 B^2 (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac {326 B^2 (b c-a d) n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}\\ \end {align*}
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Mathematica [B] time = 1.96, size = 742, normalized size = 2.42 \[ \frac {i \left (-3 B n \left (-2 a d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+2 \left (\log \left (\frac {c}{d}+x\right ) \left (-a d \log \left (\frac {d (a+b x)}{a d-b c}\right )+a d \log (a+b x)+b c\right )+(a d \log (a+b x)-b d x) \log \left (\frac {a+b x}{c+d x}\right )+a d-b c\right )+a d \log ^2\left (\frac {a}{b}+x\right )-2 a d (\log (a+b x)+1) \log \left (\frac {a}{b}+x\right )\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )+3 b B c n \left (-2 \left (\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{a d-b c}\right )\right )-2 \log (a+b x) \left (-\log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )\right )+\log ^2\left (\frac {a}{b}+x\right )\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )+3 b d x \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )^2+3 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )^2+B^2 n^2 \left (-6 a d \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+6 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (a d \log \left (\frac {a+b x}{c+d x}\right )-a d+b c\right )+\log \left (\frac {a+b x}{c+d x}\right ) \left (-a d \log ^2\left (\frac {a+b x}{c+d x}\right )+3 d \left (a \log \left (\frac {b c-a d}{b c+b d x}\right )+a+b x\right ) \log \left (\frac {a+b x}{c+d x}\right )+6 (b c-a d) \log \left (\frac {b c-a d}{b c+b d x}\right )\right )\right )-3 b B^2 c n^2 \left (-2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )\right )\right )}{3 b^2 g} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} d i x + A^{2} c i + {\left (B^{2} d i x + B^{2} c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B d i x + A B c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b g x + a g}, x\right ) \]
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 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{b g x +a g}\, dx \]
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 \[ A^{2} d i {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {A^{2} c i \log \left (b g x + a g\right )}{b g} + \frac {{\left (B^{2} b d i x + {\left (b c i - a d i\right )} B^{2} \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{b^{2} g} - \int -\frac {B^{2} b^{2} c^{2} i \log \relax (e)^{2} + 2 \, A B b^{2} c^{2} i \log \relax (e) + {\left (B^{2} b^{2} d^{2} i \log \relax (e)^{2} + 2 \, A B b^{2} d^{2} i \log \relax (e)\right )} x^{2} + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} c d i \log \relax (e)^{2} + 2 \, A B b^{2} c d i \log \relax (e)\right )} x + 2 \, {\left (B^{2} b^{2} c^{2} i \log \relax (e) + A B b^{2} c^{2} i + {\left (B^{2} b^{2} d^{2} i \log \relax (e) + A B b^{2} d^{2} i\right )} x^{2} + 2 \, {\left (B^{2} b^{2} c d i \log \relax (e) + A B b^{2} c d i\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b^{2} c^{2} i \log \relax (e) + A B b^{2} c^{2} i + {\left ({\left (i n + i \log \relax (e)\right )} B^{2} b^{2} d^{2} + A B b^{2} d^{2} i\right )} x^{2} + {\left (2 \, A B b^{2} c d i + {\left (a b d^{2} i n + 2 \, b^{2} c d i \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b^{2} c d i n - a b d^{2} i n\right )} B^{2} x + {\left (a b c d i n - a^{2} d^{2} i n\right )} B^{2}\right )} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{3} d g x^{2} + a b^{2} c g + {\left (b^{3} c g + a b^{2} d g\right )} x}\,{d x} \]
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 \[ \int \frac {\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{a\,g+b\,g\,x} \,d x \]
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 \[ \frac {i \left (\int \frac {A^{2} c}{a + b x}\, dx + \int \frac {A^{2} d x}{a + b x}\, dx + \int \frac {B^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {B^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx\right )}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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