Integrand size = 28, antiderivative size = 782 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=-\frac {n \log \left (-\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c e-b f+\sqrt {b^2-4 a c} f-c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
ln(g*(c*x^2+b*x+a)^n)*ln(e+2*f*x-(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)-n* ln(f*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(f*(b+(-4*a*c+b^2)^(1/2))-c*(e-(-4*d*f+e ^2)^(1/2))))*ln(e+2*f*x-(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)-ln(g*(c*x^2 +b*x+a)^n)*ln(e+2*f*x+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)-n*ln(e+2*f*x- (-4*d*f+e^2)^(1/2))*ln(-f*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(c*e-b*f+f*(-4*a*c+ b^2)^(1/2)-c*(-4*d*f+e^2)^(1/2)))/(-4*d*f+e^2)^(1/2)+n*ln(e+2*f*x+(-4*d*f+ e^2)^(1/2))*ln(f*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(f*(b-(-4*a*c+b^2)^(1/2))-c* (e+(-4*d*f+e^2)^(1/2))))/(-4*d*f+e^2)^(1/2)+n*ln(e+2*f*x+(-4*d*f+e^2)^(1/2 ))*ln(f*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(f*(b+(-4*a*c+b^2)^(1/2))-c*(e+(-4*d* f+e^2)^(1/2))))/(-4*d*f+e^2)^(1/2)-n*polylog(2,-c*(e+2*f*x-(-4*d*f+e^2)^(1 /2))/(f*(b-(-4*a*c+b^2)^(1/2))-c*(e-(-4*d*f+e^2)^(1/2))))/(-4*d*f+e^2)^(1/ 2)-n*polylog(2,-c*(e+2*f*x-(-4*d*f+e^2)^(1/2))/(f*(b+(-4*a*c+b^2)^(1/2))-c *(e-(-4*d*f+e^2)^(1/2))))/(-4*d*f+e^2)^(1/2)+n*polylog(2,-c*(e+2*f*x+(-4*d *f+e^2)^(1/2))/(f*(b-(-4*a*c+b^2)^(1/2))-c*(e+(-4*d*f+e^2)^(1/2))))/(-4*d* f+e^2)^(1/2)+n*polylog(2,-c*(e+2*f*x+(-4*d*f+e^2)^(1/2))/(f*(b+(-4*a*c+b^2 )^(1/2))-c*(e+(-4*d*f+e^2)^(1/2))))/(-4*d*f+e^2)^(1/2)
Time = 0.74 (sec) , antiderivative size = 663, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\frac {-n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{-c e+b f-\sqrt {b^2-4 a c} f+c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )-n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )+n \log \left (\frac {f \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )+n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )+\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )-n \operatorname {PolyLog}\left (2,\frac {c \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right )-n \operatorname {PolyLog}\left (2,\frac {c \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right )+n \operatorname {PolyLog}\left (2,\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )}\right )+n \operatorname {PolyLog}\left (2,\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{-\left (\left (b+\sqrt {b^2-4 a c}\right ) f\right )+c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
(-(n*Log[(f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(-(c*e) + b*f - Sqrt[b^2 - 4* a*c]*f + c*Sqrt[e^2 - 4*d*f])]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x]) - n*Log [(f*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + Sqrt[b^2 - 4*a*c])*f + c*(-e + Sqrt[e^2 - 4*d*f]))]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x] + n*Log[(f*(-b + S qrt[b^2 - 4*a*c] - 2*c*x))/((-b + Sqrt[b^2 - 4*a*c])*f + c*(e + Sqrt[e^2 - 4*d*f]))]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x] + n*Log[(f*(b + Sqrt[b^2 - 4 *a*c] + 2*c*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))]*L og[e + Sqrt[e^2 - 4*d*f] + 2*f*x] + Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x]*Log [g*(a + x*(b + c*x))^n] - Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x]*Log[g*(a + x* (b + c*x))^n] - n*PolyLog[2, (c*(-e + Sqrt[e^2 - 4*d*f] - 2*f*x))/((b - Sq rt[b^2 - 4*a*c])*f + c*(-e + Sqrt[e^2 - 4*d*f]))] - n*PolyLog[2, (c*(-e + Sqrt[e^2 - 4*d*f] - 2*f*x))/((b + Sqrt[b^2 - 4*a*c])*f + c*(-e + Sqrt[e^2 - 4*d*f]))] + n*PolyLog[2, (c*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/((-b + Sqrt [b^2 - 4*a*c])*f + c*(e + Sqrt[e^2 - 4*d*f]))] + n*PolyLog[2, (c*(e + Sqrt [e^2 - 4*d*f] + 2*f*x))/(-((b + Sqrt[b^2 - 4*a*c])*f) + c*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f]
Time = 1.57 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3008, 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.
| \(\displaystyle \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx\) |
| \(\Big \downarrow \) 3008 |
| \(\displaystyle \int \left (\frac {2 f \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f} \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}-\frac {2 f \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f} \left (\sqrt {e^2-4 d f}+e+2 f x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (-\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \sqrt {b^2-4 a c}-b f-c \sqrt {e^2-4 d f}+c e}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}\) |
-((n*Log[-((f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*e - b*f + Sqrt[b^2 - 4*a *c]*f - c*Sqrt[e^2 - 4*d*f]))]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^ 2 - 4*d*f]) - (n*Log[(f*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e - Sqrt[e^2 - 4*d*f]))]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x] )/Sqrt[e^2 - 4*d*f] + (n*Log[(f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((b - Sqr t[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f] + (n*Log[(f*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/( (b + Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f] + (Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x]* Log[g*(a + b*x + c*x^2)^n])/Sqrt[e^2 - 4*d*f] - (Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x]*Log[g*(a + b*x + c*x^2)^n])/Sqrt[e^2 - 4*d*f] - (n*PolyLog[2, -( (c*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/((b - Sqrt[b^2 - 4*a*c])*f - c*(e - Sq rt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] - (n*PolyLog[2, -((c*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e - Sqrt[e^2 - 4*d*f])) )])/Sqrt[e^2 - 4*d*f] + (n*PolyLog[2, -((c*(e + Sqrt[e^2 - 4*d*f] + 2*f*x) )/((b - Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4* d*f] + (n*PolyLog[2, -((c*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f]
3.1.95.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With [{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u ]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti onQ[RGx, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.78 (sec) , antiderivative size = 637, normalized size of antiderivative = 0.81
2*(ln((c*x^2+b*x+a)^n)-n*ln(c*x^2+b*x+a))/(4*d*f-e^2)^(1/2)*arctan((2*f*x+ e)/(4*d*f-e^2)^(1/2))+n*sum((ln(x-_alpha)*ln(c*x^2+b*x+a)-ln(x-_alpha)*ln( (RootOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+a*f-c*d,index =1)-x+_alpha)/RootOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+ a*f-c*d,index=1))-ln(x-_alpha)*ln((RootOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b *_alpha*f-_alpha*c*e+a*f-c*d,index=2)-x+_alpha)/RootOf(_Z^2*c*f+(2*_alpha* c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+a*f-c*d,index=2))-dilog((RootOf(_Z^2*c*f +(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+a*f-c*d,index=1)-x+_alpha)/Ro otOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+a*f-c*d,index=1) )-dilog((RootOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_alpha*c*e+a*f-c *d,index=2)-x+_alpha)/RootOf(_Z^2*c*f+(2*_alpha*c*f+b*f)*_Z+b*_alpha*f-_al pha*c*e+a*f-c*d,index=2)))/(2*_alpha*f+e),_alpha=RootOf(_Z^2*f+_Z*e+d))+2* (1/2*I*Pi*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*g*(c*x^2+b*x+a)^n)^2-1/2*I*Pi*csg n(I*(c*x^2+b*x+a)^n)*csgn(I*g*(c*x^2+b*x+a)^n)*csgn(I*g)-1/2*I*Pi*csgn(I*g *(c*x^2+b*x+a)^n)^3+1/2*I*Pi*csgn(I*g*(c*x^2+b*x+a)^n)^2*csgn(I*g)+ln(g))/ (4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))
\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d} \,d x } \]
Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
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*d*f-e^2>0)', see `assume?` for more deta
\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d} \,d x } \]
Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int \frac {\ln \left (g\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{f\,x^2+e\,x+d} \,d x \]