Integrand size = 21, antiderivative size = 130 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 a b n x}{e}+\frac {2 b^2 n^2 x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2} \]
-2*a*b*n*x/e+2*b^2*n^2*x/e-2*b^2*n*x*ln(c*x^n)/e+x*(a+b*ln(c*x^n))^2/e-d*( a+b*ln(c*x^n))^2*ln(1+e*x/d)/e^2-2*b*d*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d) /e^2+2*b^2*d*n^2*polylog(3,-e*x/d)/e^2
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {e x \left (a+b \log \left (c x^n\right )\right )^2-2 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-2 b d n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{e^2} \]
(e*x*(a + b*Log[c*x^n])^2 - 2*b*e*n*x*(a - b*n + b*Log[c*x^n]) - d*(a + b* Log[c*x^n])^2*Log[1 + (e*x)/d] - 2*b*d*n*((a + b*Log[c*x^n])*PolyLog[2, -( (e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/e^2
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 a b n x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 n^2 x}{e}\) |
(-2*a*b*n*x)/e + (2*b^2*n^2*x)/e - (2*b^2*n*x*Log[c*x^n])/e + (x*(a + b*Lo g[c*x^n])^2)/e - (d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^2 - (2*b*d*n* (a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^2 + (2*b^2*d*n^2*PolyLog[3, - ((e*x)/d)])/e^2
3.1.94.3.1 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 = 0.43 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.06
| method | result | size |
| risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} x}{e}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d \ln \left (e x +d \right )}{e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x}{e}+\frac {2 b^{2} n^{2} x}{e}-\frac {2 b^{2} d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{2}}+\frac {2 b^{2} n d \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} d \operatorname {dilog}\left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{2}}+\frac {2 b^{2} n d \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}+\frac {b^{2} d \,n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{2}}-\frac {b^{2} d \,n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} d \,n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{2}}+\frac {2 b^{2} d \,n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{2}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) x}{e}-\frac {\ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{2}}-n \left (\frac {e x +d}{e^{2}}-\frac {d \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{2}}\right )\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )}{4}\) | \(528\) |
b^2*ln(x^n)^2/e*x-b^2*ln(x^n)^2*d/e^2*ln(e*x+d)-2*b^2*n*ln(x^n)/e*x+2*b^2* n^2*x/e-2*b^2*d/e^2*ln(e*x+d)*ln(-e*x/d)*ln(x)*n^2+2*b^2*n*d/e^2*ln(x^n)*l n(e*x+d)*ln(-e*x/d)-2*b^2*d/e^2*dilog(-e*x/d)*ln(x)*n^2+2*b^2*n*d/e^2*ln(x ^n)*dilog(-e*x/d)+b^2*d/e^2*n^2*ln(e*x+d)*ln(x)^2-b^2*d/e^2*n^2*ln(x)^2*ln (1+e*x/d)-2*b^2*d/e^2*n^2*ln(x)*polylog(2,-e*x/d)+2*b^2*d*n^2*polylog(3,-e *x/d)/e^2+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*cs gn(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)*b*(ln(x^n)/e*x-ln(x^n)*d/e^2*ln(e*x+d)-n*((e*x+d)/e^2-d/e^2*( dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))))+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*(x/e-d/e^2*ln(e*x+d))
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{e x + d} \,d x } \]
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{e x + d} \,d x } \]
a^2*(x/e - d*log(e*x + d)/e^2) + integrate((b^2*x*log(x^n)^2 + 2*(b^2*log( c) + a*b)*x*log(x^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x)/(e*x + d), x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]