Optimal. Leaf size=138 \[ -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2549, 2379,
2421, 6724} \begin {gather*} \frac {2 B n \text {PolyLog}\left (2,\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 g}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b g}-\frac {\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 g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2379
Rule 2421
Rule 2549
Rule 6724
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}{a g+b g x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{a+b x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B (b c-a d) n) \int \left (\frac {d \left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b 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 (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B n) \int \left (\frac {A \log (a g+b g x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{g}-\frac {(2 B d n) \int \left (\frac {A \log (a g+b g x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 A B n) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(2 A B d n) \int \frac {\log (a g+b g x)}{-c-d x} \, dx}{b 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 (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-(2 A B n) \int \frac {\log \left (\frac {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx-\frac {(2 A B n) \text {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\frac {\left (2 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (2 B^2 d n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}+\frac {\left (B^2 n^2\right ) \int \frac {\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (B^2 d n^2\right ) \int \frac {\log ^2(a g+b g x)}{c+d x} \, dx}{b 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 (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}-\frac {(2 A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}-\frac {(2 A B n) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^n\right ) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b g}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c g+a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}+\left (2 B^2 n^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\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 {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c g+a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^n\right )}{\frac {-b c+a d}{d}-\frac {b x}{d}} \, dx,x,-c-d x\right )}{d g}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{\frac {-b c g+a d g}{d}-\frac {b g x}{d}} \, dx,x,-c-d x\right )}{d}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b g \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b 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 ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {-b c+a d}{b}-\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{b g}-\frac {d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {\left (B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {-b c+a d}{b}-\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 g}-\frac {\left (B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c g+a d g}{b g}-\frac {d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac {B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1-\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac {B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B n \log ^2(g (a+b x))}{b g}+\frac {B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac {B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 n \log \left (\frac {b (c+d x)}{b c-a d}\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 (a g+b g x)}{b g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(537\) vs. \(2(138)=276\).
time = 0.20, size = 537, normalized size = 3.89 \begin {gather*} \frac {3 \log (a+b x) \left (A+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 )\right )^2+3 B n \left (A+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 )\right ) \left (\log ^2\left (\frac {a}{b}+x\right )-2 \log (a+b x) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B^2 n^2 \left (\log ^3\left (\frac {a}{b}+x\right )+3 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+3 \log (a+b x) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {a+b x}{c+d x}\right )\right )^2+3 \log ^2\left (\frac {a}{b}+x\right ) \left (-\log \left (\frac {c}{d}+x\right )+\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+6 \log \left (\frac {a}{b}+x\right ) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )+6 \log \left (\frac {c}{d}+x\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {a}{b}+x\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-6 \text {Li}_3\left (\frac {d (a+b x)}{-b c+a d}\right )-6 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{3 b g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{b g x +a g}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {A^{2}}{a + b x}\, dx + \int \frac {B^{2} \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 \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx}{g} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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