Integrand size = 18, antiderivative size = 308 \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3} \] Output:
-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*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))*ln(x)*ln(1+2*c*x/(b-(-4* a*c+b^2)^(1/2)))/a^3-1/2*(b^2-a*c-b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))*ln(x) *ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))/a^3-1/2*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a* c+b^2)^(1/2))*polylog(2,-2*c*x/(b-(-4*a*c+b^2)^(1/2)))/a^3-1/2*(b^2-a*c-b* (-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))*polylog(2,-2*c*x/(b+(-4*a*c+b^2)^(1/2)))/ a^3
Time = 0.50 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.01 \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {\frac {a^2}{x^2}-\frac {4 a b}{x}+\frac {2 a^2 \log (x)}{x^2}-\frac {4 a b \log (x)}{x}-2 \left (b^2-a c\right ) \log ^2(x)+2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{4 a^3} \] Input:
Integrate[Log[x]/(x^3*(a + b*x + c*x^2)),x]
Output:
-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 = 1.19 (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 |
| \(\displaystyle -\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3}+\frac {\log ^2(x) \left (b^2-a c\right )}{2 a^3}-\frac {\log (x) \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a^3}-\frac {\log (x) \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a^3}+\frac {b}{a^2 x}+\frac {b \log (x)}{a^2 x}-\frac {1}{4 a x^2}-\frac {\log (x)}{2 a x^2}\) |
Input:
Int[Log[x]/(x^3*(a + b*x + c*x^2)),x]
Output:
-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)
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 = 2.20 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.54
| method | result | size |
| default | \(\frac {-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}}{a}-\frac {\left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right ) b}{a^{2}}+\frac {\frac {\ln \left (x \right ) \left (\sqrt {-4 a c +b^{2}}\, \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) a c -\sqrt {-4 a c +b^{2}}\, \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}+\sqrt {-4 a c +b^{2}}\, \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) a c -\sqrt {-4 a c +b^{2}}\, \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}+3 \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) a b c -\ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}-3 \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) a b c +\ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}\right )}{2 \sqrt {-4 a c +b^{2}}}+\frac {\sqrt {-4 a c +b^{2}}\, \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) a c -\sqrt {-4 a c +b^{2}}\, \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}+\sqrt {-4 a c +b^{2}}\, \operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) a c -\sqrt {-4 a c +b^{2}}\, \operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}+3 \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) a b c -\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}-3 \operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) a b c +\operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}}{2 \sqrt {-4 a c +b^{2}}}}{a^{3}}+\frac {\left (-a c +b^{2}\right ) \ln \left (x \right )^{2}}{2 a^{3}}\) | \(782\) |
| risch | \(-\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}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) c}{2 a^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) c}{2 a^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {3 \ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}-\frac {3 \ln \left (x \right ) \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) c}{2 a^{2}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {\operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) c}{2 a^{2}}-\frac {\operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {3 \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}-\frac {3 \operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {b +\sqrt {-4 a c +b^{2}}+2 x c}{b +\sqrt {-4 a c +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}-\frac {\ln \left (x \right )^{2} c}{2 a^{2}}+\frac {\ln \left (x \right )^{2} b^{2}}{2 a^{3}}\) | \(816\) |
Input:
int(ln(x)/x^3/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
(-1/2/x^2*ln(x)-1/4/x^2)/a-(-ln(x)/x-1/x)/a^2*b+(1/2*ln(x)*((-4*a*c+b^2)^( 1/2)*ln((-2*x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*a*c-(-4*a*c +b^2)^(1/2)*ln((-2*x*c+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^2+ (-4*a*c+b^2)^(1/2)*ln((b+(-4*a*c+b^2)^(1/2)+2*x*c)/(b+(-4*a*c+b^2)^(1/2))) *a*c-(-4*a*c+b^2)^(1/2)*ln((b+(-4*a*c+b^2)^(1/2)+2*x*c)/(b+(-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^3-3*ln((b+(- 4*a*c+b^2)^(1/2)+2*x*c)/(b+(-4*a*c+b^2)^(1/2)))*a*b*c+ln((b+(-4*a*c+b^2)^( 1/2)+2*x*c)/(b+(-4*a*c+b^2)^(1/2)))*b^3)/(-4*a*c+b^2)^(1/2)+1/2*((-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)))*a*c -(-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)))*b^2+(-4*a*c+b^2)^(1/2)*dilog((b+(-4*a*c+b^2)^(1/2)+2*x*c)/(b+(-4*a* c+b^2)^(1/2)))*a*c-(-4*a*c+b^2)^(1/2)*dilog((b+(-4*a*c+b^2)^(1/2)+2*x*c)/( b+(-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^3-3*dilog((b+(-4*a*c+b^2)^(1/2)+2*x*c)/(b+(-4*a*c+b^2)^(1/2)))* a*b*c+dilog((b+(-4*a*c+b^2)^(1/2)+2*x*c)/(b+(-4*a*c+b^2)^(1/2)))*b^3)/(-4* a*c+b^2)^(1/2))/a^3+1/2*(-a*c+b^2)*ln(x)^2/a^3
\[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int { \frac {\log \left (x\right )}{{\left (c x^{2} + b x + a\right )} x^{3}} \,d x } \] Input:
integrate(log(x)/x^3/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
integral(log(x)/(c*x^5 + b*x^4 + a*x^3), x)
Timed out. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(ln(x)/x**3/(c*x**2+b*x+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(log(x)/x^3/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
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
\[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int { \frac {\log \left (x\right )}{{\left (c x^{2} + b x + a\right )} x^{3}} \,d x } \] Input:
integrate(log(x)/x^3/(c*x^2+b*x+a),x, algorithm="giac")
Output:
integrate(log(x)/((c*x^2 + b*x + a)*x^3), x)
Timed out. \[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\int \frac {\ln \left (x\right )}{x^3\,\left (c\,x^2+b\,x+a\right )} \,d x \] Input:
int(log(x)/(x^3*(a + b*x + c*x^2)),x)
Output:
int(log(x)/(x^3*(a + b*x + c*x^2)), x)
\[ \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (x \right )}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a c \,x^{2}+4 \left (\int \frac {\mathrm {log}\left (x \right )}{c \,x^{3}+b \,x^{2}+a x}d x \right ) b^{2} x^{2}+4 \left (\int \frac {\mathrm {log}\left (x \right )}{c \,x^{2}+b x +a}d x \right ) b c \,x^{2}-2 \,\mathrm {log}\left (x \right ) a +4 \,\mathrm {log}\left (x \right ) b x -a +4 b x}{4 a^{2} x^{2}} \] Input:
int(log(x)/x^3/(c*x^2+b*x+a),x)
Output:
( - 4*int(log(x)/(a*x + b*x**2 + c*x**3),x)*a*c*x**2 + 4*int(log(x)/(a*x + b*x**2 + c*x**3),x)*b**2*x**2 + 4*int(log(x)/(a + b*x + c*x**2),x)*b*c*x* *2 - 2*log(x)*a + 4*log(x)*b*x - a + 4*b*x)/(4*a**2*x**2)