Optimal. Leaf size=831 \[ \frac{n (a+b x) \log (a+b x)}{b h}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log \left (-\frac{b \left (g+2 h x-\sqrt{g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right ) \log (a+b x)}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log \left (-\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right ) \log (a+b x)}{2 h^2}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+2 h x-\sqrt{g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 h^2}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (a+b x)}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (a+b x)}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (c+d x)}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (c+d x)}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.06817, antiderivative size = 831, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 12, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2513, 2418, 2389, 2295, 2394, 2393, 2391, 703, 634, 618, 206, 628} \[ \frac{n (a+b x) \log (a+b x)}{b h}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log \left (-\frac{b \left (g+2 h x-\sqrt{g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right ) \log (a+b x)}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log \left (-\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right ) \log (a+b x)}{2 h^2}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+2 h x-\sqrt{g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 h^2}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (a+b x)}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (a+b x)}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (c+d x)}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{PolyLog}\left (2,\frac{2 h (c+d x)}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2513
Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 703
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx &=n \int \frac{x^2 \log (a+b x)}{f+g x+h x^2} \, dx-n \int \frac{x^2 \log (c+d x)}{f+g x+h x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac{x^2}{f+g x+h x^2} \, dx\\ &=-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+n \int \left (\frac{\log (a+b x)}{h}-\frac{(f+g x) \log (a+b x)}{h \left (f+g x+h x^2\right )}\right ) \, dx-n \int \left (\frac{\log (c+d x)}{h}-\frac{(f+g x) \log (c+d x)}{h \left (f+g x+h x^2\right )}\right ) \, dx-\frac{\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac{-f-g x}{f+g x+h x^2} \, dx}{h}\\ &=-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{n \int \log (a+b x) \, dx}{h}-\frac{n \int \frac{(f+g x) \log (a+b x)}{f+g x+h x^2} \, dx}{h}-\frac{n \int \log (c+d x) \, dx}{h}+\frac{n \int \frac{(f+g x) \log (c+d x)}{f+g x+h x^2} \, dx}{h}+\frac{\left (g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac{g+2 h x}{f+g x+h x^2} \, dx}{2 h^2}-\frac{\left (\left (g^2-2 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac{1}{f+g x+h x^2} \, dx}{2 h^2}\\ &=-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^2}-\frac{n \int \left (\frac{\left (g+\frac{-g^2+2 f h}{\sqrt{g^2-4 f h}}\right ) \log (a+b x)}{g-\sqrt{g^2-4 f h}+2 h x}+\frac{\left (g-\frac{-g^2+2 f h}{\sqrt{g^2-4 f h}}\right ) \log (a+b x)}{g+\sqrt{g^2-4 f h}+2 h x}\right ) \, dx}{h}+\frac{n \int \left (\frac{\left (g+\frac{-g^2+2 f h}{\sqrt{g^2-4 f h}}\right ) \log (c+d x)}{g-\sqrt{g^2-4 f h}+2 h x}+\frac{\left (g-\frac{-g^2+2 f h}{\sqrt{g^2-4 f h}}\right ) \log (c+d x)}{g+\sqrt{g^2-4 f h}+2 h x}\right ) \, dx}{h}+\frac{n \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{b h}-\frac{n \operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{d h}+\frac{\left (\left (g^2-2 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{g^2-4 f h-x^2} \, dx,x,g+2 h x\right )}{h^2}\\ &=\frac{n (a+b x) \log (a+b x)}{b h}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^2}-\frac{\left (\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log (a+b x)}{g-\sqrt{g^2-4 f h}+2 h x} \, dx}{h}+\frac{\left (\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log (c+d x)}{g-\sqrt{g^2-4 f h}+2 h x} \, dx}{h}-\frac{\left (\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log (a+b x)}{g+\sqrt{g^2-4 f h}+2 h x} \, dx}{h}+\frac{\left (\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log (c+d x)}{g+\sqrt{g^2-4 f h}+2 h x} \, dx}{h}\\ &=\frac{n (a+b x) \log (a+b x)}{b h}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^2}+\frac{\left (b \left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log \left (\frac{b \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^2}-\frac{\left (d \left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log \left (\frac{d \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^2}+\frac{\left (b \left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log \left (\frac{b \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^2}-\frac{\left (d \left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \int \frac{\log \left (\frac{d \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^2}\\ &=\frac{n (a+b x) \log (a+b x)}{b h}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^2}+\frac{\left (\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 h x}{-2 a h+b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^2}-\frac{\left (\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 h x}{-2 c h+d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^2}+\frac{\left (\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 h x}{-2 a h+b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^2}-\frac{\left (\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 h x}{-2 c h+d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^2}\\ &=\frac{n (a+b x) \log (a+b x)}{b h}-\frac{n (c+d x) \log (c+d x)}{d h}-\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h}+\frac{\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac{g+2 h x}{\sqrt{g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2 \sqrt{g^2-4 f h}}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g-\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac{b \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac{d \left (g+\sqrt{g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{g \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^2}-\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{Li}_2\left (\frac{2 h (a+b x)}{2 a h-b \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}-\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{Li}_2\left (\frac{2 h (a+b x)}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g-\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{Li}_2\left (\frac{2 h (c+d x)}{2 c h-d \left (g-\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}+\frac{\left (g+\frac{g^2-2 f h}{\sqrt{g^2-4 f h}}\right ) n \text{Li}_2\left (\frac{2 h (c+d x)}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )}{2 h^2}\\ \end{align*}
Mathematica [A] time = 5.47177, size = 1105, normalized size = 1.33 \[ \frac{2 d h \sqrt{g^2-4 f h} (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 b d f h \log \left (g+2 h x-\sqrt{g^2-4 f h}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+b d g \left (g-\sqrt{g^2-4 f h}\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 b d f h \log \left (g+2 h x+\sqrt{g^2-4 f h}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-b d g \left (g+\sqrt{g^2-4 f h}\right ) \log \left (g+2 h x+\sqrt{g^2-4 f h}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 (b c-a d) h \sqrt{g^2-4 f h} n \log (c+d x)+2 b d f h n \left (\left (\log \left (\frac{2 h (a+b x)}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\log \left (\frac{2 h (c+d x)}{-g d+\sqrt{g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{2 c h+d \left (\sqrt{g^2-4 f h}-g\right )}\right )\right )-b d g \left (g-\sqrt{g^2-4 f h}\right ) n \left (\left (\log \left (\frac{2 h (a+b x)}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\log \left (\frac{2 h (c+d x)}{-g d+\sqrt{g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{-g b+\sqrt{g^2-4 f h} b+2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (-g-2 h x+\sqrt{g^2-4 f h}\right )}{2 c h+d \left (\sqrt{g^2-4 f h}-g\right )}\right )\right )-2 b d f h n \left (\left (\log \left (\frac{2 h (a+b x)}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )-\log \left (\frac{2 h (c+d x)}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )\right ) \log \left (g+2 h x+\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{b \left (g+\sqrt{g^2-4 f h}\right )-2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{d \left (g+\sqrt{g^2-4 f h}\right )-2 c h}\right )\right )+b d g \left (g+\sqrt{g^2-4 f h}\right ) n \left (\left (\log \left (\frac{2 h (a+b x)}{2 a h-b \left (g+\sqrt{g^2-4 f h}\right )}\right )-\log \left (\frac{2 h (c+d x)}{2 c h-d \left (g+\sqrt{g^2-4 f h}\right )}\right )\right ) \log \left (g+2 h x+\sqrt{g^2-4 f h}\right )+\text{PolyLog}\left (2,\frac{b \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{b \left (g+\sqrt{g^2-4 f h}\right )-2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (g+2 h x+\sqrt{g^2-4 f h}\right )}{d \left (g+\sqrt{g^2-4 f h}\right )-2 c h}\right )\right )}{2 b d h^2 \sqrt{g^2-4 f h}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.393, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{h{x}^{2}+gx+f}\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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