Optimal. Leaf size=55 \[ \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{B (b c-a d) f g n (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2573, 2561,
2339, 30} \begin {gather*} \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f g n (p+1) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2339
Rule 2561
Rule 2573
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx &=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{B (b c-a d) f g n (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 51, normalized size = 0.93 \begin {gather*} \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{(b B c f g n-a B d f g n) (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{p}}{\left (b f x +a f \right ) \left (d g x +c g \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 79, normalized size = 1.44 \begin {gather*} \frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )} {\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}^{p}}{{\left (B b c - B a d\right )} f g n p + {\left (B b c - B a d\right )} f g n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.35, size = 50, normalized size = 0.91 \begin {gather*} \frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}^{p + 1}}{{\left (B b c f g n - B a d f g n\right )} {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{\left (a\,f+b\,f\,x\right )\,\left (c\,g+d\,g\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________