Integrand size = 32, antiderivative size = 685 \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=-\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h \sqrt {g^2-4 f h}}+\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h}-\frac {\left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h}-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h}+\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}-\frac {\left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) n \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h} \]
-1/2*(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 +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))))*(1-g/(-4*f*h+g^2)^(1/2))/h-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))))*(1-g/(-4*f*h+g^2)^(1/2)) /h+1/2*n*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(1-g/(-4* f*h+g^2)^(1/2))/h-1/2*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g-(-4*f*h+g^2)^(1/ 2))))*(1-g/(-4*f*h+g^2)^(1/2))/h+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))))*(1+g/(-4*f*h+g^2)^(1/2))/h-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))))*(1+g/(-4*f*h+g^2)^(1/2))/h+1/2*n*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g +(-4*f*h+g^2)^(1/2))))*(1+g/(-4*f*h+g^2)^(1/2))/h-1/2*n*polylog(2,2*h*(d*x +c)/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2))))*(1+g/(-4*f*h+g^2)^(1/2))/h-g*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/(-4*f*h+g^2)^(1/2)
Time = 0.45 (sec) , antiderivative size = 539, normalized size of antiderivative = 0.79 \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\frac {\left (-g+\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\left (g+\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+\left (g-\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{2 c h+d \left (-g+\sqrt {g^2-4 f h}\right )}\right )\right )-\left (g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right )}{2 h \sqrt {g^2-4 f h}} \]
((-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] + (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] + (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*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))/(-(b*g) + 2*a*h + b*Sq rt[g^2 - 4*f*h])] - PolyLog[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f*h]))]) - (g + Sqrt[g^2 - 4*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] + PolyL og[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^2 - 4*f*h] + 2*h*x))/(-2*c*h + d*(g + Sq rt[g^2 - 4*f*h]))]))/(2*h*Sqrt[g^2 - 4*f*h])
Time = 1.12 (sec) , antiderivative size = 648, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2993, 1142, 1083, 219, 1103, 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.
| \(\displaystyle \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \frac {x}{h x^2+g x+f}dx\right )+n \int \frac {x \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \left (\frac {\int \frac {g+2 h x}{h x^2+g x+f}dx}{2 h}-\frac {g \int \frac {1}{h x^2+g x+f}dx}{2 h}\right )\right )+n \int \frac {x \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \left (\frac {g \int \frac {1}{g^2-(g+2 h x)^2-4 f h}d(g+2 h x)}{h}+\frac {\int \frac {g+2 h x}{h x^2+g x+f}dx}{2 h}\right )\right )+n \int \frac {x \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \left (\frac {\int \frac {g+2 h x}{h x^2+g x+f}dx}{2 h}+\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h \sqrt {g^2-4 f h}}\right )\right )+n \int \frac {x \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x \log (c+d x)}{h x^2+g x+f}dx\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle n \int \frac {x \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x \log (c+d x)}{h x^2+g x+f}dx-\left (\left (\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h \sqrt {g^2-4 f h}}+\frac {\log \left (f+g x+h x^2\right )}{2 h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle n \int \left (\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g+2 h x-\sqrt {g^2-4 f h}}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (a+b x)}{g+2 h x+\sqrt {g^2-4 f h}}\right )dx-n \int \left (\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g+2 h x-\sqrt {g^2-4 f h}}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (c+d x)}{g+2 h x+\sqrt {g^2-4 f h}}\right )dx-\left (\left (\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h \sqrt {g^2-4 f h}}+\frac {\log \left (f+g x+h x^2\right )}{2 h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (\left (\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h \sqrt {g^2-4 f h}}+\frac {\log \left (f+g x+h x^2\right )}{2 h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )+n \left (\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (a+b x) \log \left (-\frac {b \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (a+b x) \log \left (-\frac {b \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 h}\right )-n \left (\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (c+d x) \log \left (-\frac {d \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h}+\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (c+d x) \log \left (-\frac {d \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 h}\right )\) |
-((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*((g*A rcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]])/(h*Sqrt[g^2 - 4*f*h]) + Log[f + g*x + h*x^2]/(2*h))) + n*(((1 - g/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 ) + ((1 + g/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) + ((1 - g/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) + ((1 + g/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)) - n*(((1 - g/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) + ((1 + g/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) + ((1 - g/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - S qrt[g^2 - 4*f*h]))])/(2*h) + ((1 + g/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))
3.1.84.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]]
\[\int \frac {x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}d x\]
\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]
Timed out. \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]
Timed out. \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int \frac {x\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]