Integrand size = 18, antiderivative size = 308 \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3} \]
-1/4/a/x^2+b/a^2/x-1/2*ln(x)/a/x^2+b*ln(x)/a^2/x+1/2*(-a*c+b^2)*ln(x)^2/a^ 3-1/2*ln(x)*ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))*(b^2-a*c-b*(-3*a*c+b^2)/(-4 *a*c+b^2)^(1/2))/a^3-1/2*polylog(2,-2*c*x/(b+(-4*a*c+b^2)^(1/2)))*(b^2-a*c -b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^3-1/2*ln(x)*ln(1+2*c*x/(b-(-4*a*c+b^ 2)^(1/2)))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^3-1/2*polylog(2,- 2*c*x/(b-(-4*a*c+b^2)^(1/2)))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/ a^3
Time = 0.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.01 \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {\frac {a^2}{x^2}-\frac {4 a b}{x}+\frac {2 a^2 \log (x)}{x^2}-\frac {4 a b \log (x)}{x}-2 \left (b^2-a c\right ) \log ^2(x)+2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{4 a^3} \]
-1/4*(a^2/x^2 - (4*a*b)/x + (2*a^2*Log[x])/x^2 - (4*a*b*Log[x])/x - 2*(b^2 - a*c)*Log[x]^2 + 2*(b^2 - a*c + (b*(b^2 - 3*a*c))/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])] + 2*(b^2 - a*c - (b*(b^2 - 3*a*c))/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])] + 2*(b^2 - a*c + (b*(b^2 - 3*a*c))/ Sqrt[b^2 - 4*a*c])*PolyLog[2, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[ b^2 - 4*a*c])])/a^3
Time = 0.64 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
| \(\displaystyle \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \int \left (\frac {\log (x) \left (-c x \left (b^2-a c\right )-b \left (b^2-2 a c\right )\right )}{a^3 \left (a+b x+c x^2\right )}+\frac {\log (x) \left (b^2-a c\right )}{a^3 x}-\frac {b \log (x)}{a^2 x^2}+\frac {\log (x)}{a x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3}+\frac {\log ^2(x) \left (b^2-a c\right )}{2 a^3}-\frac {\log (x) \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a^3}-\frac {\log (x) \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a^3}+\frac {b}{a^2 x}+\frac {b \log (x)}{a^2 x}-\frac {1}{4 a x^2}-\frac {\log (x)}{2 a x^2}\) |
-1/4*1/(a*x^2) + b/(a^2*x) - Log[x]/(2*a*x^2) + (b*Log[x])/(a^2*x) + ((b^2 - a*c)*Log[x]^2)/(2*a^3) - ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a *c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a* c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[ b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c ])*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c - ( b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4* a*c])])/(2*a^3)
3.4.57.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(781\) vs. \(2(278)=556\).
Time = 1.16 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.54
| method | result | size |
| default | \(\frac {\left (-c a +b^{2}\right ) \ln \left (x \right )^{2}}{2 a^{3}}+\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}\right )}{2 \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 \sqrt {-4 c a +b^{2}}}}{a^{3}}+\frac {-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}}{a}-\frac {b \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{a^{2}}\) | \(782\) |
| risch | \(-\frac {\ln \left (x \right )^{2} c}{2 a^{2}}+\frac {\ln \left (x \right )^{2} b^{2}}{2 a^{3}}+\frac {\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 ) c}{2 a^{2}}-\frac {\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 ) b^{2}}{2 a^{3}}+\frac {3 \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 ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}-\frac {\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 ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\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 ) c}{2 a^{2}}-\frac {\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 ) b^{2}}{2 a^{3}}-\frac {3 \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 ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\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 ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {3 \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}-\frac {3 \operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right )}{2 a \,x^{2}}-\frac {1}{4 a \,x^{2}}+\frac {b \ln \left (x \right )}{a^{2} x}+\frac {b}{a^{2} x}\) | \(816\) |
1/2*(-a*c+b^2)*ln(x)^2/a^3+1/a^3*(1/2*ln(x)*(ln((-2*x*c+(-4*a*c+b^2)^(1/2) -b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-ln((-2*x*c+(-4*a*c+b^2 )^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2+3*ln((-2*x*c+(- 4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*a*b*c-ln((-2*x*c+(-4*a*c+b^2) ^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3+ln((2*x*c+(-4*a*c+b^2)^(1/2)+b)/(b+ (-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-ln((2*x*c+(-4*a*c+b^2)^(1/2)+b )/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2-3*ln((2*x*c+(-4*a*c+b^2)^ (1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*a*b*c+ln((2*x*c+(-4*a*c+b^2)^(1/2)+b)/(b+ (-4*a*c+b^2)^(1/2)))*b^3)/(-4*a*c+b^2)^(1/2)+1/2*(dilog((-2*x*c+(-4*a*c+b^ 2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-dilog((-2*x*c+ (-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2+3*di log((-2*x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*a*b*c-dilog((-2 *x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3+dilog((2*x*c+(-4*a *c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-dilog((2*x *c+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2-3* dilog((2*x*c+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*a*b*c+dilog((2* x*c+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b^3)/(-4*a*c+b^2)^(1/2)) +1/a*(-1/2/x^2*ln(x)-1/4/x^2)-1/a^2*b*(-1/x*ln(x)-1/x)
\[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int { \frac {\log \left (x\right )}{{\left (c x^{2} + b x + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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*a*c-b^2>0)', see `assume?` for more deta
\[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int { \frac {\log \left (x\right )}{{\left (c x^{2} + b x + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int \frac {\ln \left (x\right )}{x^3\,\left (c\,x^2+b\,x+a\right )} \,d x \]