Optimal. Leaf size=148 \[ -\frac{B n \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{B n \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{\log (g+h x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}-\frac{B n \log (g+h x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h}+\frac{B n \log (g+h x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.188928, antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {6742, 2494, 2394, 2393, 2391} \[ -\frac{B n \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{B n \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{B \log (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac{B n \log (g+h x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h}+\frac{A \log (g+h x)}{h}+\frac{B n \log (g+h x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2494
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx &=\int \left (\frac{A}{g+h x}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}\right ) \, dx\\ &=\frac{A \log (g+h x)}{h}+B \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx\\ &=\frac{A \log (g+h x)}{h}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}-\frac{(b B n) \int \frac{\log (g+h x)}{a+b x} \, dx}{h}+\frac{(B d n) \int \frac{\log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac{A \log (g+h x)}{h}-\frac{B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+(B n) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx-(B n) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx\\ &=\frac{A \log (g+h x)}{h}-\frac{B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{(B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}-\frac{(B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}\\ &=\frac{A \log (g+h x)}{h}-\frac{B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}-\frac{B n \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{B n \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}\\ \end{align*}
Mathematica [A] time = 0.0949328, size = 150, normalized size = 1.01 \[ \frac{B n \left (\text{PolyLog}\left (2,\frac{h (a+b x)}{a h-b g}\right )+\log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )\right )-B n \left (\text{PolyLog}\left (2,\frac{h (c+d x)}{c h-d g}\right )+\log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )\right )+\log (g+h x) \left (B \left (\log \left (e (a+b x)^n (c+d x)^{-n}\right )-n \log (a+b x)+n \log (c+d x)\right )+A\right )}{h} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.783, size = 597, normalized size = 4. \begin{align*} \text{result too large to display} \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*} -B \int -\frac{\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \left (e\right )}{h x + g}\,{d x} + \frac{A \log \left (h x + g\right )}{h} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g}, x\right ) \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{B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]