Optimal. Leaf size=151 \[ -\frac {B^2 i n^2 (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {B i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2561, 2342,
2341} \begin {gather*} -\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B^2 i n^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2561
Rubi steps
\begin {align*} \int \frac {(165 c+165 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)^3} \, dx &=\int \left (\frac {165 (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^3 (a+b x)^3}+\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(165 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)^2} \, dx}{b g^3}+\frac {(165 (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)^3} \, dx}{b g^3}\\ &=-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(330 B d 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 )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(165 B (b c-a d) 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 )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(330 B d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (165 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(330 B d (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 )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (165 B (b c-a d)^2 n\right ) \int \left (\frac {b \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)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}-\frac {(165 B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(330 B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {\left (165 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (330 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (165 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (330 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {(165 B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (330 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac {\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac {\left (165 B^2 d^2 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^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 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 (b c-a d) g^3}-\frac {\left (330 B^2 d^2 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^2 (b c-a d) g^3}+\frac {\left (165 B^2 (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac {\left (165 B^2 d^2 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^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 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 (b c-a d) g^3}-\frac {\left (330 B^2 d^2 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^2 (b c-a d) g^3}-\frac {\left (165 B^2 d (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (330 B^2 d (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (165 B^2 (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (165 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (330 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (165 B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (330 B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (330 B^2 d (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (165 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}\\ &=-\frac {165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac {165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac {165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (165 B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac {165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac {165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {165 B^2 d^2 n^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {165 B^2 d^2 n^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (165 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (330 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}\\ &=-\frac {165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac {165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac {165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {165 B^2 d^2 n^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {165 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {165 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {165 B d^2 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 (b c-a d) g^3}-\frac {165 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {165 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {165 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {165 B^2 d^2 n^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {165 B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.60, size = 801, normalized size = 5.30 \begin {gather*} -\frac {i \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-4 d (-b c+a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+4 B d n (a+b x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+2 B n (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d n (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+B d n (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B n \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B n \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 n (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 b^2 (b c-a d) g^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2013 vs. \(2 (144) = 288\).
time = 0.41, size = 2013, normalized size = 13.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 478 vs. \(2 (144) = 288\).
time = 0.37, size = 478, normalized size = 3.17 \begin {gather*} \frac {2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b^{2} c^{2} + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a^{2} d^{2} - {\left (i \, B^{2} b^{2} c^{2} - i \, B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left (-i \, B^{2} b^{2} d^{2} n^{2} x^{2} - 2 i \, B^{2} b^{2} c d n^{2} x - i \, B^{2} b^{2} c^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{2} c^{2} + {\left (i \, A B + i \, B^{2}\right )} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b^{2} c d + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b d^{2} + {\left (-i \, B^{2} b^{2} c d + i \, B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{2} c d + {\left (i \, A B + i \, B^{2}\right )} a b d^{2}\right )} n\right )} x + 2 \, {\left (-i \, B^{2} b^{2} c^{2} n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} c^{2} n + {\left (-i \, B^{2} b^{2} d^{2} n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (-i \, B^{2} b^{2} c d n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} c d n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \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 {i \left (\int \frac {A^{2} c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {A^{2} d x}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, 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^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, 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^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, 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^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, 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^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx\right )}{g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.45, size = 177, normalized size = 1.17 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 i \, {\left (d x + c\right )}^{2} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )}^{2} g^{3}} + \frac {2 \, {\left (i \, B^{2} n^{2} + 2 i \, A B n + 2 i \, B^{2} n\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{2} g^{3}} + \frac {{\left (i \, B^{2} n^{2} + 2 i \, A B n + 2 i \, B^{2} n + 2 i \, A^{2} + 4 i \, A B + 2 i \, B^{2}\right )} {\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.82, size = 561, normalized size = 3.72 \begin {gather*} -{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b}+\frac {B^2\,d\,i\,x}{b}+\frac {B^2\,a\,d\,i}{2\,b^2}}{a^2\,g^3+2\,a\,b\,g^3\,x+b^2\,g^3\,x^2}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B\,a\,d\,i+A\,B\,b\,c\,i-B^2\,a\,d\,i\,n+B^2\,b\,c\,i\,n+2\,A\,B\,b\,d\,i\,x}{a^2\,b^2\,g^3+2\,a\,b^3\,g^3\,x+b^4\,g^3\,x^2}+\frac {B^2\,d^2\,i\,\left (\frac {a\,b^2\,g^3\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^3\,n\,x\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,g^3\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}\right )}{b^2\,g^3\,\left (a\,d-b\,c\right )\,\left (a^2\,b^2\,g^3+2\,a\,b^3\,g^3\,x+b^4\,g^3\,x^2\right )}\right )-\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B\,n+b\,d\,i\,B^2\,n^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i\,n^2}{2}+\frac {B^2\,b\,c\,i\,n^2}{2}+A\,B\,a\,d\,i\,n+A\,B\,b\,c\,i\,n}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {B\,d^2\,i\,n\,\mathrm {atan}\left (\frac {B\,d^2\,i\,n\,\left (2\,A+B\,n\right )\,\left (\frac {c\,b^3\,g^3+a\,d\,b^2\,g^3}{b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (i\,B^2\,d^2\,n^2+2\,A\,i\,B\,d^2\,n\right )}\right )\,\left (2\,A+B\,n\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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