Integrand size = 19, antiderivative size = 129 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=-n \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right ) \]
ln(x)*ln(d*(c*x^2+b*x+a)^n)-n*ln(x)*ln(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))-n*l n(x)*ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))-n*polylog(2,-2*c*x/(b-(-4*a*c+b^2) ^(1/2)))-n*polylog(2,-2*c*x/(b+(-4*a*c+b^2)^(1/2)))
Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\log (x) \log \left (d (a+x (b+c x))^n\right )-n \left (\log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+\operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )+\operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )\right ) \]
Log[x]*Log[d*(a + x*(b + c*x))^n] - n*(Log[x]*Log[(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c])] + Log[x]*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c *x)/(b + Sqrt[b^2 - 4*a*c])] + PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c]) ] + PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])
Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3004, 2804, 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.
\(\displaystyle \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 3004 |
\(\displaystyle \log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \int \frac {(b+2 c x) \log (x)}{c x^2+b x+a}dx\) |
\(\Big \downarrow \) 2804 |
\(\displaystyle \log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \int \left (\frac {2 c \log (x)}{b+2 c x-\sqrt {b^2-4 a c}}+\frac {2 c \log (x)}{b+2 c x+\sqrt {b^2-4 a c}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \left (\operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )+\operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )+\log (x) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )\right )\) |
Log[x]*Log[d*(a + b*x + c*x^2)^n] - n*(Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^ 2 - 4*a*c])] + Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])] + PolyLog[2 , (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])] + PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])
3.1.76.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[Log[d + e*x]*((a + b*Log[c*RFx^p])^n/e), x] - Simp[b*n*(p/e) Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29
method | result | size |
parts | \(\ln \left (x \right ) \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )-n \left (\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )+\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )+\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )+\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )\right )\) | \(166\) |
risch | \(\ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right ) \ln \left (x \right )-\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \ln \left (x \right ) n -\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \ln \left (x \right ) n -\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) n -\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) n +\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i d \right )}{2}+\ln \left (d \right )\right ) \ln \left (x \right )\) | \(308\) |
ln(x)*ln(d*(c*x^2+b*x+a)^n)-n*(ln(x)*ln((-2*x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+ (-4*a*c+b^2)^(1/2)))+ln(x)*ln((2*x*c+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2) ^(1/2)))+dilog((-2*x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))+dilo g((2*x*c+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2))))
\[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x} \,d x } \]
\[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\int \frac {\log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{x}\, dx \]
\[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x} \,d x } \]
\[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx=\int \frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{x} \,d x \]