Integrand size = 26, antiderivative size = 939 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {b^2 e^2 m n^2 \log (x)}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x)}{2 d^2}+\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right )}{d^2}-\frac {3 b^2 e^2 m n^2 \log (d+e x)}{2 d^2}-\frac {3 b^2 e m n^2 \log (d+e x)}{2 d x}+\frac {b^2 e^2 m n^2 \log (x) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(x) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log (d+e x)}{d^2}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log (d+e x)}{d x}-\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(d+e x)}{4 d^2}-\frac {b^2 m n^2 \log ^2(d+e x)}{4 x^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{2 d^2}+\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 d^2}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 x^2}+\frac {b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d^2 x^2}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{2 x^2}-\frac {\left (m-2 m \log (x)+2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{4 x^2}-\frac {b^2 e^2 m n^2 \log (x) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 n^2 \left (m-\log \left (f x^m\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}-\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )}{2 d^2 x^2}-\frac {b^2 e^2 m n^2 (1+2 \log (d+e x)) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^2}-\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{d^2} \]
b^2*e^2*m*n^2*ln(x)*ln(e*x+d)/d^2-b^2*e^2*n^2*ln(x)*ln(f*x^m)*ln(e*x+d)/d^ 2-b^2*e^2*m*n^2*ln(x)*ln(1+e*x/d)/d^2+b^2*e^2*n^2*ln(x)*ln(f*x^m)*ln(1+e*x /d)/d^2-b^2*e*n^2*ln(f*x^m)*ln(e*x+d)/d/x+b*n*(m*ln(x)-ln(f*x^m))*(e^2*x^2 *ln(-e*x/d)+(e*x+d)*(e*x+(-e*x+d)*ln(e*x+d)))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x +d)^n))/d^2/x^2-1/4*(m-2*m*ln(x)+2*ln(f*x^m))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x +d)^n))^2/x^2-3/2*b^2*e*m*n^2*ln(e*x+d)/d/x+1/2*b^2*e^2*m*n^2*ln(x)^2*ln(e *x+d)/d^2-1/2*b^2*e^2*m*n^2*ln(-e*x/d)*ln(e*x+d)/d^2-1/2*b^2*e^2*m*n^2*ln( -e*x/d)*ln(e*x+d)^2/d^2-1/2*b^2*e^2*m*n^2*ln(x)^2*ln(1+e*x/d)/d^2-1/2*b*m* n*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(e*x*(e*x+d)+e^2*x^2*ln(-e*x/d)+(-e^ 2*x^2+d^2)*ln(e*x+d)+2*d^2*ln(x)*ln(e*x+d)+e*x*(e*x*ln(x)^2+2*d*(1+ln(x))- 2*e*x*(ln(x)*ln(1+e*x/d)+polylog(2,-e*x/d))))/d^2/x^2-1/2*b^2*e^2*m*n^2*(1 +2*ln(e*x+d))*polylog(2,1+e*x/d)/d^2-1/4*b^2*m*n^2*ln(e*x+d)^2/x^2-1/2*b^2 *n^2*ln(f*x^m)*ln(e*x+d)^2/x^2-1/2*m*ln(x)*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d) ^n))^2/x^2-1/2*b^2*e^2*m*n^2*ln(x)^2/d^2+1/2*b^2*e^2*m*n^2*ln(-e*x/d)/d^2- 3/2*b^2*e^2*m*n^2*ln(e*x+d)/d^2+1/4*b^2*e^2*m*n^2*ln(e*x+d)^2/d^2+1/2*b^2* e^2*n^2*ln(f*x^m)*ln(e*x+d)^2/d^2-b^2*e^2*m*n^2*polylog(3,-e*x/d)/d^2+b^2* e^2*m*n^2*polylog(3,1+e*x/d)/d^2-b^2*e^2*n^2*ln(f*x^m)*ln(e*x+d)/d^2+b^2*e ^2*m*n^2*ln(x)/d^2+b^2*e^2*n^2*ln(x)*ln(f*x^m)/d^2-b^2*e^2*n^2*(m-ln(f*x^m ))*polylog(2,-e*x/d)/d^2
Time = 0.62 (sec) , antiderivative size = 781, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {4 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-2 d^2 m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d^2 \left (-m+2 m \log (x)-2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (4 e^2 m x^2 \log (x)-2 e^2 m x^2 \log ^2(x)+2 e^2 m x^2 \log \left (-\frac {e x}{d}\right )+4 e^2 x^2 \log (x) \log \left (f x^m\right )-6 d e m x \log (d+e x)-6 e^2 m x^2 \log (d+e x)+4 e^2 m x^2 \log (x) \log (d+e x)-2 e^2 m x^2 \log ^2(x) \log (d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)+4 e^2 m x^2 \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)-4 d e x \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-d^2 m \log ^2(d+e x)+e^2 m x^2 \log ^2(d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-2 d^2 \log \left (f x^m\right ) \log ^2(d+e x)+2 e^2 x^2 \log \left (f x^m\right ) \log ^2(d+e x)-4 e^2 m x^2 \log (x) \log \left (1+\frac {e x}{d}\right )+2 e^2 m x^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )+4 e^2 m x^2 (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 e^2 x^2 \left (m-2 m \log (x)+2 \log \left (f x^m\right )+2 m \log (d+e x)\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-4 e^2 m x^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+4 e^2 m x^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )}{4 d^2 x^2} \]
(4*b*n*(m*Log[x] - Log[f*x^m])*(e^2*x^2*Log[-((e*x)/d)] + (d + e*x)*(e*x + (d - e*x)*Log[d + e*x]))*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]) - 2*d^2*m*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + d^2*(-m + 2*m*Log[x] - 2*Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^ 2 - 2*b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e*x*(d + e*x) + e^2*x^2*Log[-((e*x)/d)] + (d^2 - e^2*x^2)*Log[d + e*x] + 2*d^2*Log[x]*Log [d + e*x] + e*x*(e*x*Log[x]^2 + 2*d*(1 + Log[x]) - 2*e*x*(Log[x]*Log[1 + ( e*x)/d] + PolyLog[2, -((e*x)/d)]))) + b^2*n^2*(4*e^2*m*x^2*Log[x] - 2*e^2* m*x^2*Log[x]^2 + 2*e^2*m*x^2*Log[-((e*x)/d)] + 4*e^2*x^2*Log[x]*Log[f*x^m] - 6*d*e*m*x*Log[d + e*x] - 6*e^2*m*x^2*Log[d + e*x] + 4*e^2*m*x^2*Log[x]* Log[d + e*x] - 2*e^2*m*x^2*Log[x]^2*Log[d + e*x] - 2*e^2*m*x^2*Log[-((e*x) /d)]*Log[d + e*x] + 4*e^2*m*x^2*Log[x]*Log[-((e*x)/d)]*Log[d + e*x] - 4*d* e*x*Log[f*x^m]*Log[d + e*x] - 4*e^2*x^2*Log[f*x^m]*Log[d + e*x] - 4*e^2*x^ 2*Log[-((e*x)/d)]*Log[f*x^m]*Log[d + e*x] - d^2*m*Log[d + e*x]^2 + e^2*m*x ^2*Log[d + e*x]^2 - 2*e^2*m*x^2*Log[-((e*x)/d)]*Log[d + e*x]^2 - 2*d^2*Log [f*x^m]*Log[d + e*x]^2 + 2*e^2*x^2*Log[f*x^m]*Log[d + e*x]^2 - 4*e^2*m*x^2 *Log[x]*Log[1 + (e*x)/d] + 2*e^2*m*x^2*Log[x]^2*Log[1 + (e*x)/d] + 4*e^2*m *x^2*(-1 + Log[x])*PolyLog[2, -((e*x)/d)] - 2*e^2*x^2*(m - 2*m*Log[x] + 2* Log[f*x^m] + 2*m*Log[d + e*x])*PolyLog[2, 1 + (e*x)/d] - 4*e^2*m*x^2*PolyL og[3, -((e*x)/d)] + 4*e^2*m*x^2*PolyLog[3, 1 + (e*x)/d]))/(4*d^2*x^2)
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.
| \(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 2876 |
| \(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3}dx\) |
3.4.72.3.1 Defintions of rubi rules used
Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_ .))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> Unintegrable[(g*x)^q*Log[f*x^m]* (a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q} , x]
\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{3}}d x\]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
integral((b^2*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*log((e*x + d)^n*c)*l og(f*x^m) + a^2*log(f*x^m))/x^3, x)
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
-1/4*(b^2*(m + 2*log(f)) + 2*b^2*log(x^m))*log((e*x + d)^n)^2/x^2 + integr ate(1/2*(2*b^2*d*log(c)^2*log(f) + 4*a*b*d*log(c)*log(f) + 2*a^2*d*log(f) + 2*(b^2*e*log(c)^2*log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log(f))*x + (4* b^2*d*log(c)*log(f) + 4*a*b*d*log(f) + (4*a*b*e*log(f) + (4*e*log(c)*log(f ) + (m*n + 2*n*log(f))*e)*b^2)*x + 2*(2*b^2*d*log(c) + 2*a*b*d + ((e*n + 2 *e*log(c))*b^2 + 2*a*b*e)*x)*log(x^m))*log((e*x + d)^n) + 2*(b^2*d*log(c)^ 2 + 2*a*b*d*log(c) + a^2*d + (b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x)* log(x^m))/(e*x^4 + d*x^3), x)
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^3} \,d x \]