Integrand size = 23, antiderivative size = 135 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=-\frac {2 b^2 n^2}{d x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}-\frac {2 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2} \]
-2*b^2*n^2/d/x-2*b*n*(a+b*ln(c*x^n))/d/x-(a+b*ln(c*x^n))^2/d/x+e*ln(1+d/e/ x)*(a+b*ln(c*x^n))^2/d^2-2*b*e*n*(a+b*ln(c*x^n))*polylog(2,-d/e/x)/d^2-2*b ^2*e*n^2*polylog(3,-d/e/x)/d^2
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=-\frac {\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+\frac {6 b d n \left (a+b n+b \log \left (c x^n\right )\right )}{x}-3 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-6 b e n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{3 d^2} \]
-1/3*((3*d*(a + b*Log[c*x^n])^2)/x + (e*(a + b*Log[c*x^n])^3)/(b*n) + (6*b *d*n*(a + b*n + b*Log[c*x^n]))/x - 3*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/ d] - 6*b*e*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, - ((e*x)/d)]))/d^2
Time = 0.76 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2780, 2742, 2741, 2779, 2821, 7143}
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{x^2}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\) |
(-((a + b*Log[c*x^n])^2/x) + 2*b*n*(-((b*n)/x) - (a + b*Log[c*x^n])/x))/d - (e*(-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d) + (2*b*n*((a + b*Log[c* x^n])*PolyLog[2, -(d/(e*x))] + b*n*PolyLog[3, -(d/(e*x))]))/d))/d
3.1.97.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* (x_)^(r_.)), x_Symbol] :> Simp[1/d Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Simp[e/d Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.41 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.56
| method | result | size |
| risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} e \ln \left (e x +d \right )}{d^{2}}-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{d x}-\frac {b^{2} \ln \left (x^{n}\right )^{2} e \ln \left (x \right )}{d^{2}}+\frac {2 b^{2} e \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}+\frac {2 b^{2} e \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}-\frac {2 b^{2} n e \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n e \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b^{2} e \,n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{2}}+\frac {b^{2} e \,n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{2}}+\frac {2 b^{2} e \,n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} e \,n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right )}{d x}-\frac {2 b^{2} n^{2}}{d x}+\frac {b^{2} n e \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{2}}-\frac {b^{2} e \ln \left (x \right )^{3} n^{2}}{3 d^{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 ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (x^{n}\right )}{d x}-\frac {\ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{2}}-n \left (\frac {e \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{d^{2}}+\frac {1}{d x}-\frac {e \ln \left (x \right )^{2}}{2 d^{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 {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )}{4}\) | \(615\) |
b^2*ln(x^n)^2*e/d^2*ln(e*x+d)-b^2*ln(x^n)^2/d/x-b^2*ln(x^n)^2*e/d^2*ln(x)+ 2*b^2*e/d^2*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2+2*b^2*e/d^2*ln(x)*dilog(-e*x/d) *n^2-2*b^2*n*e/d^2*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-2*b^2*n*e/d^2*ln(x^n)*dilo g(-e*x/d)-b^2*e/d^2*n^2*ln(e*x+d)*ln(x)^2+b^2*e/d^2*n^2*ln(x)^2*ln(1+e*x/d )+2*b^2*e/d^2*n^2*ln(x)*polylog(2,-e*x/d)-2*b^2*e/d^2*n^2*polylog(3,-e*x/d )-2*b^2*n*ln(x^n)/d/x-2*b^2*n^2/d/x+b^2*n*e/d^2*ln(x^n)*ln(x)^2-1/3*b^2*e/ d^2*ln(x)^3*n^2+(-I*b*Pi*csgn(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*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/d^2*ln(e*x+d)-ln(x^n)/d/x-ln(x^n)*e/d^2*ln (x)-n*(e/d^2*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))+1/d/x-1/2*e/d^2*ln(x)^2) )+1/4*(-I*b*Pi*csgn(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*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*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
a^2*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + integrate((b^2*log(c)^ 2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e*x^3 + d*x^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,\left (d+e\,x\right )} \,d x \]