Integrand size = 22, antiderivative size = 711 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}} \]
3/4*b*n*(a+b*ln(c*x^n))^2*ln(1-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-1/ 4*(a+b*ln(c*x^n))^3*ln(1-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3/4*b*n* (a+b*ln(c*x^n))^2*ln(1+x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+1/4*(a+b*l n(c*x^n))^3*ln(1+x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3/2*b^2*n^2*(a+b *ln(c*x^n))*polylog(2,-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+3/4*b*n*(a +b*ln(c*x^n))^2*polylog(2,-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+3/2*b^ 2*n^2*(a+b*ln(c*x^n))*polylog(2,x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3 /4*b*n*(a+b*ln(c*x^n))^2*polylog(2,x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2 )+3/2*b^3*n^3*polylog(3,-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3/2*b^2* n^2*(a+b*ln(c*x^n))*polylog(3,-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3/ 2*b^3*n^3*polylog(3,x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+3/2*b^2*n^2*( a+b*ln(c*x^n))*polylog(3,x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+3/2*b^3* n^3*polylog(4,-x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)-3/2*b^3*n^3*polylo g(4,x*e^(1/2)/(-d)^(1/2))/(-d)^(3/2)/e^(1/2)+1/4*x*(a+b*ln(c*x^n))^3/(-d)^ (3/2)/((-d)^(1/2)-x*e^(1/2))+1/4*x*(a+b*ln(c*x^n))^3/(-d)^(3/2)/((-d)^(1/2 )+x*e^(1/2))
Result contains complex when optimal does not.
Time = 2.91 (sec) , antiderivative size = 1073, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {d} x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3}{d+e x^2}+\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3}{\sqrt {e}}+3 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\frac {\sqrt {e} x \log (x)+i \left (\sqrt {d}+i \sqrt {e} x\right ) \log \left (i \sqrt {d}-\sqrt {e} x\right )}{\sqrt {d} \sqrt {e}+i e x}+\frac {\sqrt {e} x \log (x)+\left (-i \sqrt {d}-\sqrt {e} x\right ) \log \left (i \sqrt {d}+\sqrt {e} x\right )}{\sqrt {d} \sqrt {e}-i e x}-\frac {i \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e}}+\frac {i \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e}}\right )+3 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac {\log (x) \left (\sqrt {e} x \log (x)+2 i \left (\sqrt {d}+i \sqrt {e} x\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+2 i \left (\sqrt {d}+i \sqrt {e} x\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}+i e x}+\frac {\log (x) \left (\sqrt {e} x \log (x)-2 i \left (\sqrt {d}-i \sqrt {e} x\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-2 \left (i \sqrt {d}+\sqrt {e} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}-i e x}-\frac {i \left (\log ^2(x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \log (x) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e}}+\frac {i \left (\log ^2(x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \log (x) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )-2 \operatorname {PolyLog}\left (3,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e}}\right )+\frac {i b^3 n^3 \left (-\log ^3(x)+\frac {\sqrt {d} \log ^3(x)}{\sqrt {d}+i \sqrt {e} x}+\frac {\sqrt {e} x \log ^3(x)}{i \sqrt {d}+\sqrt {e} x}-3 \log ^2(x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\log ^3(x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+3 \log ^2(x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\log ^3(x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )-3 (-2+\log (x)) \log (x) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+3 (-2+\log (x)) \log (x) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )-6 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+6 \log (x) \operatorname {PolyLog}\left (3,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+6 \operatorname {PolyLog}\left (3,\frac {i \sqrt {e} x}{\sqrt {d}}\right )-6 \log (x) \operatorname {PolyLog}\left (3,\frac {i \sqrt {e} x}{\sqrt {d}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+6 \operatorname {PolyLog}\left (4,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e}}}{4 d^{3/2}} \]
((2*Sqrt[d]*x*(a - b*n*Log[x] + b*Log[c*x^n])^3)/(d + e*x^2) + (2*ArcTan[( Sqrt[e]*x)/Sqrt[d]]*(a - b*n*Log[x] + b*Log[c*x^n])^3)/Sqrt[e] + 3*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*((Sqrt[e]*x*Log[x] + I*(Sqrt[d] + I*Sqrt[e] *x)*Log[I*Sqrt[d] - Sqrt[e]*x])/(Sqrt[d]*Sqrt[e] + I*e*x) + (Sqrt[e]*x*Log [x] + ((-I)*Sqrt[d] - Sqrt[e]*x)*Log[I*Sqrt[d] + Sqrt[e]*x])/(Sqrt[d]*Sqrt [e] - I*e*x) - (I*(Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + PolyLog[2, ((-I )*Sqrt[e]*x)/Sqrt[d]]))/Sqrt[e] + (I*(Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d] ] + PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]))/Sqrt[e]) + 3*b^2*n^2*(a - b*n*Log[ x] + b*Log[c*x^n])*((Log[x]*(Sqrt[e]*x*Log[x] + (2*I)*(Sqrt[d] + I*Sqrt[e] *x)*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]]) + (2*I)*(Sqrt[d] + I*Sqrt[e]*x)*PolyLo g[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e] + I*e*x) + (Log[x]*(Sqrt[ e]*x*Log[x] - (2*I)*(Sqrt[d] - I*Sqrt[e]*x)*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] ) - 2*(I*Sqrt[d] + Sqrt[e]*x)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]* Sqrt[e] - I*e*x) - (I*(Log[x]^2*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + 2*Log[x]* PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - 2*PolyLog[3, ((-I)*Sqrt[e]*x)/Sqrt[ d]]))/Sqrt[e] + (I*(Log[x]^2*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 2*Log[x]*Pol yLog[2, (I*Sqrt[e]*x)/Sqrt[d]] - 2*PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]]))/Sqr t[e]) + (I*b^3*n^3*(-Log[x]^3 + (Sqrt[d]*Log[x]^3)/(Sqrt[d] + I*Sqrt[e]*x) + (Sqrt[e]*x*Log[x]^3)/(I*Sqrt[d] + Sqrt[e]*x) - 3*Log[x]^2*Log[1 - (I*Sq rt[e]*x)/Sqrt[d]] + Log[x]^3*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 3*Log[x]^...
Time = 1.10 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2767, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 d \left (-d e-e^2 x^2\right )}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}-\frac {3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}+\frac {3 b n \log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}-\frac {3 b n \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt {e}}+\frac {\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}\) |
(x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b *Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (3*b*n*(a + b*Log[ c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b* Log[c*x^n])^3*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (3*b *n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[ e]) + ((a + b*Log[c*x^n])^3*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*S qrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d]) ])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt [e]*x)/Sqrt[-d])])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])* PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*L og[c*x^n])^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + (3 *b^3*n^3*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3* b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3 /2)*Sqrt[e]) - (3*b^3*n^3*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)* Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]]) /(2*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[4, -((Sqrt[e]*x)/Sqrt[-d])])/ (2*(-d)^(3/2)*Sqrt[e]) - (3*b^3*n^3*PolyLog[4, (Sqrt[e]*x)/Sqrt[-d]])/(2*( -d)^(3/2)*Sqrt[e])
3.3.48.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x ^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{\left (e \,x^{2}+d \right )^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a ^3)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (e\,x^2+d\right )}^2} \,d x \]