Integrand size = 23, antiderivative size = 211 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=-\frac {2 b^2 n^2}{d^2 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}+\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac {2 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {4 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3}-\frac {2 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {4 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3} \]
-2*b^2*n^2/d^2/x-2*b*n*(a+b*ln(c*x^n))/d^2/x-(a+b*ln(c*x^n))^2/d^2/x+e^2*x *(a+b*ln(c*x^n))^2/d^3/(e*x+d)+2*e*ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d^3-2*b*e *n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^3-4*b*e*n*(a+b*ln(c*x^n))*polylog(2,-d/e/ x)/d^3-2*b^2*e*n^2*polylog(2,-e*x/d)/d^3-4*b^2*e*n^2*polylog(3,-d/e/x)/d^3
Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=-\frac {\frac {6 b^2 d n^2}{x}+\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-3 e \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {3 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 d^3} \]
-1/3*((6*b^2*d*n^2)/x + (6*b*d*n*(a + b*Log[c*x^n]))/x - 3*e*(a + b*Log[c* x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/x + (3*d*e*(a + b*Log[c*x^n])^2)/(d + e*x) + (2*e*(a + b*Log[c*x^n])^3)/(b*n) + 6*b*e*n*(a + b*Log[c*x^n])*Log[ 1 + (e*x)/d] - 6*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 6*b^2*e*n^2*Pol yLog[2, -((e*x)/d)] - 12*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 12*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^3
Time = 0.54 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2795, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)^2}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}-\frac {4 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac {2 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {2 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {2 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {4 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3}-\frac {2 b^2 n^2}{d^2 x}\) |
(-2*b^2*n^2)/(d^2*x) - (2*b*n*(a + b*Log[c*x^n]))/(d^2*x) - (a + b*Log[c*x ^n])^2/(d^2*x) + (e^2*x*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)) + (2*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^3 - (2*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 - (4*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 - (2*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^3 - (4*b^2*e*n^2*PolyLog[3, -(d/( e*x))])/d^3
3.2.5.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.44 (sec) , antiderivative size = 790, normalized size of antiderivative = 3.74
-b^2*ln(x^n)^2/d^2*e/(e*x+d)+2*b^2*ln(x^n)^2/d^3*e*ln(e*x+d)-b^2*ln(x^n)^2 /d^2/x-2*b^2*ln(x^n)^2/d^3*e*ln(x)-2*b^2*n*ln(x^n)/d^3*e*ln(e*x+d)-2*b^2*n *ln(x^n)/d^2/x+2*b^2*n*ln(x^n)/d^3*e*ln(x)-b^2/d^3*n^2*e*ln(x)^2-2*b^2*n^2 /d^2/x+2*b^2/d^3*n^2*e*ln(e*x+d)*ln(-e*x/d)+2*b^2/d^3*n^2*e*dilog(-e*x/d)+ 2*b^2*n/d^3*e*ln(x^n)*ln(x)^2-2/3*b^2/d^3*e*ln(x)^3*n^2+4*b^2/d^3*e*ln(x)* ln(e*x+d)*ln(-e*x/d)*n^2+4*b^2/d^3*e*ln(x)*dilog(-e*x/d)*n^2-4*b^2*n/d^3*e *ln(x^n)*ln(e*x+d)*ln(-e*x/d)-4*b^2*n/d^3*e*ln(x^n)*dilog(-e*x/d)-2*b^2/d^ 3*e*n^2*ln(e*x+d)*ln(x)^2+2*b^2/d^3*e*n^2*ln(x)^2*ln(1+e*x/d)+4*b^2/d^3*e* n^2*ln(x)*polylog(2,-e*x/d)-4*b^2/d^3*e*n^2*polylog(3,-e*x/d)+(-I*b*Pi*csg n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*c sgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-ln(x^ n)/d^2*e/(e*x+d)+2*ln(x^n)/d^3*e*ln(e*x+d)-ln(x^n)/d^2/x-2*ln(x^n)/d^3*e*l n(x)-n*(-1/d^3*e*ln(x)^2+2/d^3*e*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))+1/d^ 3*e*ln(e*x+d)+1/d^2/x-1/d^3*e*ln(x)))+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*c sgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c* x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(-1/d^2*e/(e*x+d)+2/d^3*e*l n(e*x+d)-1/d^2/x-2/d^3*e*ln(x))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{2}} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{2}} \,d x } \]
-a^2*((2*e*x + d)/(d^2*e*x^2 + d^3*x) - 2*e*log(e*x + d)/d^3 + 2*e*log(x)/ d^3) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*lo g(c) + a*b)*log(x^n))/(e^2*x^4 + 2*d*e*x^3 + d^2*x^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,{\left (d+e\,x\right )}^2} \,d x \]