Optimal. Leaf size=52 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106263, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {6686} \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6686
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{a c f+(b c+a d) f x+b d f x^2} \, 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 n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0116366, size = 50, normalized size = 0.96 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{f (p+1) (b B c n-a B d n)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.884, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{acf+ \left ( ad+bc \right ) fx+bdf{x}^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{p}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f +{\left (b c + a d\right )} f x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.546326, size = 209, normalized size = 4.02 \begin{align*} \frac{{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}^{p}}{{\left (B b c - B a d\right )} f n p +{\left (B b c - B a d\right )} f n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f +{\left (b c + a d\right )} f x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]