Optimal. Leaf size=44 \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6684} \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6684
Rubi steps
\begin {align*} \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx &=\frac {\log \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B (b c-a d) f n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 42, normalized size = 0.95 \[ \frac {\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{f (b B c n-a B d n)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.98, size = 48, normalized size = 1.09 \[ \frac {\log \left (-B n \log \left (b x + a\right ) + B n \log \left (d x + c\right ) - B \log \relax (e) - A\right )}{{\left (B b c - B a d\right )} f n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 40, normalized size = 0.91 \[ \frac {\log \left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}{B b c f n - B a d f n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.21, size = 371, normalized size = 8.43 \[ -\frac {\ln \left (\ln \left (\left (d x +c \right )^{n}\right )-\frac {-i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i \pi B \,\mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+2 B \ln \relax (e )+2 B \ln \left (\left (b x +a \right )^{n}\right )+2 A}{2 B}\right )}{\left (a d -b c \right ) B f n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.58, size = 51, normalized size = 1.16 \[ \frac {\log \left (-\frac {B \log \left ({\left (b x + a\right )}^{n}\right ) - B \log \left ({\left (d x + c\right )}^{n}\right ) + B \log \relax (e) + A}{B}\right )}{{\left (b c f n - a d f n\right )} B} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.51, size = 42, normalized size = 0.95 \[ -\frac {\ln \left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}{B\,a\,d\,f\,n-B\,b\,c\,f\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________