3.4.71 \(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))^2}{x^2} \, dx\) [371]

3.4.71.1 Optimal result

Integrand size = 26, antiderivative size = 607 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=-\frac {b^2 e m n^2 \log ^2(x) \log (d+e x)}{d}+\frac {2 b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d}-\frac {b^2 e m n^2 \log ^2(d+e x)}{d}-\frac {b^2 m n^2 \log ^2(d+e x)}{x}+\frac {b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{d}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{d}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{x}-\frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e x \log \left (-\frac {e x}{d}\right )-(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d x}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}-\frac {\left (m-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}+\frac {b^2 e m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d}+\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (2 e x \log \left (-\frac {e x}{d}\right )-2 (d+e x) \log (d+e x)-2 d \log (x) \log (d+e x)+e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )}{d x}+\frac {2 b^2 e m n^2 (1+\log (d+e x)) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d}+\frac {2 b^2 e m n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d}-\frac {2 b^2 e m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{d} \]

output

-b^2*e*m*n^2*ln(x)^2*ln(e*x+d)/d+2*b^2*e*m*n^2*ln(-e*x/d)*ln(e*x+d)/d+2*b^ 
2*e*n^2*ln(x)*ln(f*x^m)*ln(e*x+d)/d-b^2*e*m*n^2*ln(e*x+d)^2/d-b^2*m*n^2*ln 
(e*x+d)^2/x+b^2*e*m*n^2*ln(-e*x/d)*ln(e*x+d)^2/d-b^2*e*n^2*ln(f*x^m)*ln(e* 
x+d)^2/d-b^2*n^2*ln(f*x^m)*ln(e*x+d)^2/x-2*b*n*(m*ln(x)-ln(f*x^m))*(e*x*ln 
(-e*x/d)-(e*x+d)*ln(e*x+d))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))/d/x-m*ln(x 
)*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2/x-(m-m*ln(x)+ln(f*x^m))*(a-b*n*ln( 
e*x+d)+b*ln(c*(e*x+d)^n))^2/x+b^2*e*m*n^2*ln(x)^2*ln(1+e*x/d)/d-2*b^2*e*n^ 
2*ln(x)*ln(f*x^m)*ln(1+e*x/d)/d-2*b^2*e*n^2*ln(f*x^m)*polylog(2,-e*x/d)/d+ 
b*m*n*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(2*e*x*ln(-e*x/d)-2*(e*x+d)*ln(e 
*x+d)-2*d*ln(x)*ln(e*x+d)+e*x*(ln(x)^2-2*ln(x)*ln(1+e*x/d)-2*polylog(2,-e* 
x/d)))/d/x+2*b^2*e*m*n^2*(1+ln(e*x+d))*polylog(2,1+e*x/d)/d+2*b^2*e*m*n^2* 
polylog(3,-e*x/d)/d-2*b^2*e*m*n^2*polylog(3,1+e*x/d)/d
 

3.4.71.2 Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (-e x \log \left (-\frac {e x}{d}\right )+(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-d m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d \left (-m+m \log (x)-\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-2 e x \log \left (-\frac {e x}{d}\right )+2 (d+e x) \log (d+e x)+2 d \log (x) \log (d+e x)-e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (e m x \log ^2(x) \log (d+e x)+2 e m x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 e m x \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e x \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-d m \log ^2(d+e x)-e m x \log ^2(d+e x)+e m x \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-d \log \left (f x^m\right ) \log ^2(d+e x)-e x \log \left (f x^m\right ) \log ^2(d+e x)-e m x \log ^2(x) \log \left (1+\frac {e x}{d}\right )-2 e m x \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+2 e x \left (m-m \log (x)+\log \left (f x^m\right )+m \log (d+e x)\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+2 e m x \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-2 e m x \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )}{d x} \]

input

Integrate[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/x^2,x]
 

output

(2*b*n*(m*Log[x] - Log[f*x^m])*(-(e*x*Log[-((e*x)/d)]) + (d + e*x)*Log[d + 
 e*x])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]) - d*m*Log[x]*(a - b*n 
*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + d*(-m + m*Log[x] - Log[f*x^m])*( 
a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - b*m*n*(a - b*n*Log[d + e* 
x] + b*Log[c*(d + e*x)^n])*(-2*e*x*Log[-((e*x)/d)] + 2*(d + e*x)*Log[d + e 
*x] + 2*d*Log[x]*Log[d + e*x] - e*x*(Log[x]^2 - 2*(Log[x]*Log[1 + (e*x)/d] 
 + PolyLog[2, -((e*x)/d)]))) + b^2*n^2*(e*m*x*Log[x]^2*Log[d + e*x] + 2*e* 
m*x*Log[-((e*x)/d)]*Log[d + e*x] - 2*e*m*x*Log[x]*Log[-((e*x)/d)]*Log[d + 
e*x] + 2*e*x*Log[-((e*x)/d)]*Log[f*x^m]*Log[d + e*x] - d*m*Log[d + e*x]^2 
- e*m*x*Log[d + e*x]^2 + e*m*x*Log[-((e*x)/d)]*Log[d + e*x]^2 - d*Log[f*x^ 
m]*Log[d + e*x]^2 - e*x*Log[f*x^m]*Log[d + e*x]^2 - e*m*x*Log[x]^2*Log[1 + 
 (e*x)/d] - 2*e*m*x*Log[x]*PolyLog[2, -((e*x)/d)] + 2*e*x*(m - m*Log[x] + 
Log[f*x^m] + m*Log[d + e*x])*PolyLog[2, 1 + (e*x)/d] + 2*e*m*x*PolyLog[3, 
-((e*x)/d)] - 2*e*m*x*PolyLog[3, 1 + (e*x)/d]))/(d*x)
 

3.4.71.3 Rubi [F]

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.

\(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx\)

\(\Big \downarrow \) 2876

\(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2}dx\)

input

Int[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/x^2,x]
 

output

$Aborted
 

3.4.71.3.1 Defintions of rubi rules used

rule 2876

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]
 

3.4.71.4 Maple [F]

\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{2}}d x\]

input

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/x^2,x)
 

output

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/x^2,x)
 

3.4.71.5 Fricas [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x^2,x, algorithm="fricas")
 

output

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^2, x)
 

3.4.71.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\text {Timed out} \]

input

integrate(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**2/x**2,x)
 

output

Timed out
 

3.4.71.7 Maxima [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x^2,x, algorithm="maxima")
 

output

-(b^2*(m + log(f)) + b^2*log(x^m))*log((e*x + d)^n)^2/x + integrate((b^2*d 
*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f) + (b^2*e*log(c)^2* 
log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log(f))*x + 2*(b^2*d*log(c)*log(f) 
+ a*b*d*log(f) + (a*b*e*log(f) + (e*log(c)*log(f) + (m*n + n*log(f))*e)*b^ 
2)*x + (b^2*d*log(c) + a*b*d + ((e*n + e*log(c))*b^2 + a*b*e)*x)*log(x^m)) 
*log((e*x + d)^n) + (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^3 + d*x^2), x)
 

3.4.71.8 Giac [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x^2,x, algorithm="giac")
 

output

integrate((b*log((e*x + d)^n*c) + a)^2*log(f*x^m)/x^2, x)
 

3.4.71.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2} \,d x \]

input

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2)/x^2,x)
 

output

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2)/x^2, x)