Integrand size = 31, antiderivative size = 97 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {B n}{b (a+b x)}-\frac {B d n \log (a+b x)}{b (b c-a d)}+\frac {B d n \log (c+d x)}{b (b c-a d)}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 46} \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{b (a+b x)}-\frac {B d n \log (a+b x)}{b (b c-a d)}+\frac {B d n \log (c+d x)}{b (b c-a d)}-\frac {B n}{b (a+b x)} \]
[In]
[Out]
Rule 46
Rule 2548
Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b} \\ & = -\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)}+\frac {(B (b c-a d) n) \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} \\ & = -\frac {B n}{b (a+b x)}-\frac {B d n \log (a+b x)}{b (b c-a d)}+\frac {B d n \log (c+d x)}{b (b c-a d)}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=\frac {-B d n (a+b x) \log (a+b x)+B d n (a+b x) \log (c+d x)-(b c-a d) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b (b c-a d) (a+b x)} \]
[In]
[Out]
Time = 5.72 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.34
method | result | size |
parallelrisch | \(-\frac {-B x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} d^{2} n -B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} c d n +B a \,b^{2} d^{2} n^{2}-B \,b^{3} c d \,n^{2}+A a \,b^{2} d^{2} n -A \,b^{3} c d n}{\left (b x +a \right ) b^{3} d n \left (a d -c b \right )}\) | \(130\) |
risch | \(\frac {B \ln \left (\left (d x +c \right )^{n}\right )}{b \left (b x +a \right )}-\frac {2 A b c -2 B a d n +2 B b c n -2 A a d -2 B a d \ln \left (\left (b x +a \right )^{n}\right )-2 B \ln \left (e \right ) a d -2 B \ln \left (d x +c \right ) a d n +2 B \ln \left (-b x -a \right ) a d n +i B \pi a d \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i B \pi a d \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )-i B \,\operatorname {csgn}\left (i e \right ) \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) b c -i B \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) b c +2 B \ln \left (-b x -a \right ) b d n x -2 B \ln \left (d x +c \right ) b d n x -i B \pi b c \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}-i B \pi b c \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}+i B \pi a d \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i B \pi a d \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}+2 B \ln \left (e \right ) b c +2 B \ln \left (\left (b x +a \right )^{n}\right ) b c -i B \pi a d \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi a d \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi a d \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi a d \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i B \pi b c \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi b c \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi b c \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i B \pi b c \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}}{2 \left (b x +a \right ) b \left (-a d +c b \right )}\) | \(823\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {A b c - A a d + {\left (B b c - B a d\right )} n + {\left (B b d n x + B b c n\right )} \log \left (b x + a\right ) - {\left (B b d n x + B b c n\right )} \log \left (d x + c\right ) + {\left (B b c - B a d\right )} \log \left (e\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {{\left (\frac {d e n \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d e n \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {e n}{b^{2} x + a b}\right )} B}{e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{b^{2} x + a b} - \frac {A}{b^{2} x + a b} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {B d n \log \left (b x + a\right )}{b^{2} c - a b d} + \frac {B d n \log \left (d x + c\right )}{b^{2} c - a b d} - \frac {B n \log \left (b x + a\right )}{b^{2} x + a b} + \frac {B n \log \left (d x + c\right )}{b^{2} x + a b} - \frac {B n + B \log \left (e\right ) + A}{b^{2} x + a b} \]
[In]
[Out]
Time = 1.52 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx=-\frac {A+B\,n}{x\,b^2+a\,b}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{b\,\left (a+b\,x\right )}-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \]
[In]
[Out]