Integrand size = 34, antiderivative size = 831 \[ \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=\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 ) \text {arctanh}\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \]
n*(b*x+a)*ln(b*x+a)/b/h-n*(d*x+c)*ln(d*x+c)/d/h-x*(n*ln(b*x+a)-ln(e*((b*x+ a)/(d*x+c))^n)-n*ln(d*x+c))/h+1/2*g*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n) -n*ln(d*x+c))*ln(h*x^2+g*x+f)/h^2-1/2*n*ln(b*x+a)*ln(-b*(g+2*h*x-(-4*f*h+g ^2)^(1/2))/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g+(2*f*h-g^2)/(-4*f*h+g^2)^( 1/2))/h^2+1/2*n*ln(d*x+c)*ln(-d*(g+2*h*x-(-4*f*h+g^2)^(1/2))/(2*c*h-d*(g-( -4*f*h+g^2)^(1/2))))*(g+(2*f*h-g^2)/(-4*f*h+g^2)^(1/2))/h^2-1/2*n*polylog( 2,2*h*(b*x+a)/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g+(2*f*h-g^2)/(-4*f*h+g^2 )^(1/2))/h^2+1/2*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g-(-4*f*h+g^2)^(1/2)))) *(g+(2*f*h-g^2)/(-4*f*h+g^2)^(1/2))/h^2-1/2*n*ln(b*x+a)*ln(-b*(g+2*h*x+(-4 *f*h+g^2)^(1/2))/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))*(g+(-2*f*h+g^2)/(-4*f*h +g^2)^(1/2))/h^2+1/2*n*ln(d*x+c)*ln(-d*(g+2*h*x+(-4*f*h+g^2)^(1/2))/(2*c*h -d*(g+(-4*f*h+g^2)^(1/2))))*(g+(-2*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^2-1/2*n* polylog(2,2*h*(b*x+a)/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))*(g+(-2*f*h+g^2)/(- 4*f*h+g^2)^(1/2))/h^2+1/2*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g+(-4*f*h+g^2) ^(1/2))))*(g+(-2*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^2+(-2*f*h+g^2)*arctanh((2* h*x+g)/(-4*f*h+g^2)^(1/2))*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x +c))/h^2/(-4*f*h+g^2)^(1/2)
Time = 0.85 (sec) , antiderivative size = 1105, normalized size of antiderivative = 1.33 \[ \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=\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 c-a d) h \sqrt {g^2-4 f h} n \log (c+d x)-2 b d f h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+b d g \left (g-\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+2 b d f h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )-b d g \left (g+\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+2 b d f h n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\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 )\right )-b d g \left (g-\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\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 )\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+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\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 )-\operatorname {PolyLog}\left (2,\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 )\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+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\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 )-\operatorname {PolyLog}\left (2,\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 )\right )}{2 b d h^2 \sqrt {g^2-4 f h}} \]
(2*d*h*Sqrt[g^2 - 4*f*h]*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*(b*c - a*d)*h*Sqrt[g^2 - 4*f*h]*n*Log[c + d*x] - 2*b*d*f*h*Log[e*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + b*d*g*(g - Sqrt[g^2 - 4*f *h])*Log[e*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + 2 *b*d*f*h*Log[e*((a + b*x)/(c + d*x))^n]*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x] - b*d*g*(g + Sqrt[g^2 - 4*f*h])*Log[e*((a + b*x)/(c + d*x))^n]*Log[g + Sq rt[g^2 - 4*f*h] + 2*h*x] + 2*b*d*f*h*n*((Log[(2*h*(a + b*x))/(-(b*g) + 2*a *h + b*Sqrt[g^2 - 4*f*h])] - Log[(2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sqrt[ g^2 - 4*f*h])])*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(-g + S qrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - PolyL og[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f *h]))]) - b*d*g*(g - Sqrt[g^2 - 4*f*h])*n*((Log[(2*h*(a + b*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - Log[(2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sq rt[g^2 - 4*f*h])])*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - Po lyLog[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f*h]))]) - 2*b*d*f*h*n*((Log[(2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - Log[(2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])*Log[ g + Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(g + Sqrt[g^2 - 4*f*h] + 2* h*x))/(-2*a*h + b*(g + Sqrt[g^2 - 4*f*h]))] - PolyLog[2, (d*(g + Sqrt[g...
Time = 1.46 (sec) , antiderivative size = 766, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2993, 1143, 2009, 2865, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \frac {x^2}{h x^2+g x+f}dx\right )+n \int \frac {x^2 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^2 \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \left (\frac {1}{h}-\frac {f+g x}{h \left (h x^2+g x+f\right )}\right )dx\right )+n \int \frac {x^2 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^2 \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle n \int \frac {x^2 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^2 \log (c+d x)}{h x^2+g x+f}dx-\left (\left (-\frac {\left (g^2-2 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^2 \sqrt {g^2-4 f h}}-\frac {g \log \left (f+g x+h x^2\right )}{2 h^2}+\frac {x}{h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle n \int \left (\frac {\log (a+b x)}{h}-\frac {(f+g x) \log (a+b x)}{h \left (h x^2+g x+f\right )}\right )dx-n \int \left (\frac {\log (c+d x)}{h}-\frac {(f+g x) \log (c+d x)}{h \left (h x^2+g x+f\right )}\right )dx-\left (\left (-\frac {\left (g^2-2 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^2 \sqrt {g^2-4 f h}}-\frac {g \log \left (f+g x+h x^2\right )}{2 h^2}+\frac {x}{h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (\left (-\frac {\left (g^2-2 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^2 \sqrt {g^2-4 f h}}-\frac {g \log \left (f+g x+h x^2\right )}{2 h^2}+\frac {x}{h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )+n \left (-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}+g\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) \log (a+b x) \log \left (-\frac {b \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}+g\right ) \log (a+b x) \log \left (-\frac {b \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 h^2}+\frac {(a+b x) \log (a+b x)}{b h}-\frac {x}{h}\right )-n \left (-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}+g\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) \log (c+d x) \log \left (-\frac {d \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^2}-\frac {\left (\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}+g\right ) \log (c+d x) \log \left (-\frac {d \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 h^2}+\frac {(c+d x) \log (c+d x)}{d h}-\frac {x}{h}\right )\) |
-((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*(x/h - ((g^2 - 2*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]])/(h^2*Sqrt[g^2 - 4 *f*h]) - (g*Log[f + g*x + h*x^2])/(2*h^2))) + n*(-(x/h) + ((a + b*x)*Log[a + b*x])/(b*h) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*Log[a + b*x]*Log[- ((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))] )/(2*h^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*Log[a + b*x]*Log[-((b*( g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2* h^2) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*(a + b*x))/( 2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^ 2 - 4*f*h])*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])) ])/(2*h^2)) - n*(-(x/h) + ((c + d*x)*Log[c + d*x])/(d*h) - ((g - (g^2 - 2* f*h)/Sqrt[g^2 - 4*f*h])*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h *x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^2) - ((g + (g^2 - 2*f*h)/ Sqrt[g^2 - 4*f*h])*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/ (2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^2) - ((g - (g^2 - 2*f*h)/Sqrt[ g^2 - 4*f*h])*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h] ))])/(2*h^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^2))
3.1.83.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r Int[RFx*Log[a + b*x], x], x] + (Si mp[q*r Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) Int[RFx, x], x]) /; FreeQ[ {a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a *d, 0] && !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ m, n]]
\[\int \frac {x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}d x\]
\[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*f*h-g^2>0)', see `assume?` for more deta
\[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int \frac {x^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]