Integrand size = 23, antiderivative size = 204 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=-\frac {b^2 n^2}{4 d x^2}+\frac {2 b^2 e n^2}{d^2 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3} \]
-1/4*b^2*n^2/d/x^2+2*b^2*e*n^2/d^2/x-1/2*b*n*(a+b*ln(c*x^n))/d/x^2+2*b*e*n *(a+b*ln(c*x^n))/d^2/x-1/2*(a+b*ln(c*x^n))^2/d/x^2+e*(a+b*ln(c*x^n))^2/d^2 /x-e^2*ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d^3+2*b*e^2*n*(a+b*ln(c*x^n))*polylog (2,-d/e/x)/d^3+2*b^2*e^2*n^2*polylog(3,-d/e/x)/d^3
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\frac {-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {12 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+\frac {24 b d e n \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {3 b d^2 n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-12 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-24 b e^2 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{12 d^3} \]
((-6*d^2*(a + b*Log[c*x^n])^2)/x^2 + (12*d*e*(a + b*Log[c*x^n])^2)/x + (4* e^2*(a + b*Log[c*x^n])^3)/(b*n) + (24*b*d*e*n*(a + b*n + b*Log[c*x^n]))/x - (3*b*d^2*n*(2*a + b*n + 2*b*Log[c*x^n]))/x^2 - 12*e^2*(a + b*Log[c*x^n]) ^2*Log[1 + (e*x)/d] - 24*b*e^2*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d) ] - b*n*PolyLog[3, -((e*x)/d)]))/(12*d^3)
Time = 1.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2780, 2742, 2741, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {b n \int \frac {a+b \log \left (c x^n\right )}{x^3}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{x^2}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\) |
(-1/2*(a + b*Log[c*x^n])^2/x^2 + b*n*(-1/4*(b*n)/x^2 - (a + b*Log[c*x^n])/ (2*x^2)))/d - (e*((-((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))/d
3.1.98.3.1 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 = 0.46 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.58
| method | result | size |
| risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 d \,x^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2} \ln \left (x \right )}{d^{3}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e}{d^{2} x}+\frac {2 b^{2} n \ln \left (x^{n}\right ) e}{d^{2} x}-\frac {b^{2} n \ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {2 b^{2} e \,n^{2}}{d^{2} x}-\frac {b^{2} n^{2}}{4 d \,x^{2}}-\frac {b^{2} n \,e^{2} \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{3}}+\frac {b^{2} e^{2} \ln \left (x \right )^{3} n^{2}}{3 d^{3}}-\frac {2 b^{2} e^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{3}}-\frac {2 b^{2} e^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{3}}+\frac {b^{2} e^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{3}}-\frac {b^{2} e^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{3}}-\frac {2 b^{2} e^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{3}}+\frac {2 b^{2} e^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{3}}+\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^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {\ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (x^{n}\right ) e}{d^{2} x}-\frac {n \left (\frac {-\frac {2 e}{x}+\frac {d}{2 x^{2}}}{d^{2}}+\frac {e^{2} \ln \left (x \right )^{2}}{d^{3}}-\frac {2 e^{2} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{d^{3}}\right )}{2}\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^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )}{4}\) | \(731\) |
-b^2*ln(x^n)^2*e^2/d^3*ln(e*x+d)-1/2*b^2*ln(x^n)^2/d/x^2+b^2*ln(x^n)^2*e^2 /d^3*ln(x)+b^2*ln(x^n)^2*e/d^2/x+2*b^2*n*ln(x^n)*e/d^2/x-1/2*b^2*n*ln(x^n) /d/x^2+2*b^2*e*n^2/d^2/x-1/4*b^2*n^2/d/x^2-b^2*n*e^2/d^3*ln(x^n)*ln(x)^2+1 /3*b^2*e^2/d^3*ln(x)^3*n^2-2*b^2*e^2/d^3*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-2* b^2*e^2/d^3*ln(x)*dilog(-e*x/d)*n^2+2*b^2*n*e^2/d^3*ln(x^n)*ln(e*x+d)*ln(- e*x/d)+2*b^2*n*e^2/d^3*ln(x^n)*dilog(-e*x/d)+b^2*e^2/d^3*n^2*ln(e*x+d)*ln( x)^2-b^2*e^2/d^3*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2*e^2/d^3*n^2*ln(x)*polylog(2 ,-e*x/d)+2*b^2*e^2/d^3*n^2*polylog(3,-e*x/d)+(-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^2/d^3*ln(e*x +d)-1/2*ln(x^n)/d/x^2+ln(x^n)*e^2/d^3*ln(x)+ln(x^n)*e/d^2/x-1/2*n*(1/d^2*( -2*e/x+1/2*d/x^2)+e^2/d^3*ln(x)^2-2*e^2/d^3*(dilog(-e*x/d)+ln(e*x+d)*ln(-e *x/d))))+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^2/d^3*ln(e*x+d)-1/2/d/x^2+e^2/d^3*ln(x)+e/d^2/x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{3} \left (d + e x\right )}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{3}} \,d x } \]
-1/2*a^2*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - d)/(d^2*x^2 )) + 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^4 + d*x^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3\,\left (d+e\,x\right )} \,d x \]