\(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))^2}{x} \, dx\) [370]

Optimal result

Integrand size = 26, antiderivative size = 823 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx =\text {Too large to display} \] Output:

1/2*m*ln(x)^2*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2+ln(x)*(-m*ln(x)+ln(f*x 
^m))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2+2*b*n*(-m*ln(x)+ln(f*x^m))*(a-b 
*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(ln(x)*(ln(e*x+d)-ln(1+e*x/d))-polylog(2,- 
e*x/d))+2*b*m*n*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(1/2*ln(x)^2*(ln(e*x+d 
)-ln(1+e*x/d))-ln(x)*polylog(2,-e*x/d)+polylog(3,-e*x/d))-b^2*n^2*(m*ln(x) 
-ln(f*x^m))*(ln(-e*x/d)*ln(e*x+d)^2+2*ln(e*x+d)*polylog(2,1+e*x/d)-2*polyl 
og(3,1+e*x/d))+1/12*b^2*m*n^2*(ln(-e*x/d)^4+6*ln(-e*x/d)^2*ln(-e*x/(e*x+d) 
)^2-4*(ln(-e*x/d)+ln(d/(e*x+d)))*ln(-e*x/(e*x+d))^3+ln(-e*x/(e*x+d))^4+6*l 
n(x)^2*ln(e*x+d)^2+4*(2*ln(-e*x/d)^3-3*ln(x)^2*ln(e*x+d))*ln(1+e*x/d)+6*(l 
n(x)-ln(-e*x/d))*(ln(x)+3*ln(-e*x/d))*ln(1+e*x/d)^2-4*ln(-e*x/d)^2*ln(-e*x 
/(e*x+d))*(ln(-e*x/d)+3*ln(1+e*x/d))+12*(ln(-e*x/d)^2-2*ln(-e*x/d)*(ln(-e* 
x/(e*x+d))+ln(1+e*x/d))+2*ln(x)*(-ln(e*x+d)+ln(1+e*x/d)))*polylog(2,-e*x/d 
)-12*ln(-e*x/(e*x+d))^2*polylog(2,e*x/(e*x+d))+12*(ln(-e*x/d)-ln(-e*x/(e*x 
+d)))^2*polylog(2,1+e*x/d)+24*(ln(x)-ln(-e*x/d))*ln(1+e*x/d)*polylog(2,1+e 
*x/d)+24*(ln(-e*x/(e*x+d))+ln(e*x+d))*polylog(3,-e*x/d)+24*ln(-e*x/(e*x+d) 
)*polylog(3,e*x/(e*x+d))+24*(-ln(x)+ln(-e*x/(e*x+d)))*polylog(3,1+e*x/d)-2 
4*polylog(4,-e*x/d)-24*polylog(4,e*x/(e*x+d))+24*polylog(4,1+e*x/d))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

(m*Log[x]^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/2 + Log[x]*(- 
(m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 
 2*b*n*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x 
)^n])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) 
+ 2*b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((Log[x]^2*(Log[d 
+ e*x] - Log[1 + (e*x)/d]))/2 - Log[x]*PolyLog[2, -((e*x)/d)] + PolyLog[3, 
 -((e*x)/d)]) - b^2*n^2*(m*Log[x] - Log[f*x^m])*(Log[-((e*x)/d)]*Log[d + e 
*x]^2 + 2*Log[d + e*x]*PolyLog[2, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d] 
) + (b^2*m*n^2*(Log[-((e*x)/d)]^4 + 6*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e 
*x))]^2 - 4*(Log[-((e*x)/d)] + Log[d/(d + e*x)])*Log[-((e*x)/(d + e*x))]^3 
 + Log[-((e*x)/(d + e*x))]^4 + 6*Log[x]^2*Log[d + e*x]^2 + 4*(2*Log[-((e*x 
)/d)]^3 - 3*Log[x]^2*Log[d + e*x])*Log[1 + (e*x)/d] + 6*(Log[x] - Log[-((e 
*x)/d)])*(Log[x] + 3*Log[-((e*x)/d)])*Log[1 + (e*x)/d]^2 - 4*Log[-((e*x)/d 
)]^2*Log[-((e*x)/(d + e*x))]*(Log[-((e*x)/d)] + 3*Log[1 + (e*x)/d]) + 12*( 
Log[-((e*x)/d)]^2 - 2*Log[-((e*x)/d)]*(Log[-((e*x)/(d + e*x))] + Log[1 + ( 
e*x)/d]) + 2*Log[x]*(-Log[d + e*x] + Log[1 + (e*x)/d]))*PolyLog[2, -((e*x) 
/d)] - 12*Log[-((e*x)/(d + e*x))]^2*PolyLog[2, (e*x)/(d + e*x)] + 12*(Log[ 
-((e*x)/d)] - Log[-((e*x)/(d + e*x))])^2*PolyLog[2, 1 + (e*x)/d] + 24*(Log 
[x] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*PolyLog[2, 1 + (e*x)/d] + 24*(Log[ 
-((e*x)/(d + e*x))] + Log[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((...
 

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.

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

\(\Big \downarrow \) 2874

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

\(\Big \downarrow \) 2891

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2874

Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[f*x^m]^2*((a + b*Log[c*(d + e*x)^n] 
)^p/(2*m)), x] - Simp[b*e*n*(p/(2*m))   Int[Log[f*x^m]^2*((a + b*Log[c*(d + 
 e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] 
&& IGtQ[p, 0]
 

rule 2891

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))^(q_.)*((k_.) + (l_.)*(x_))^(r_.), 
 x_Symbol] :> Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^p*(f + g* 
Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, 
n, p, q, r}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*(b^2*m*log(x)^2 - 2*b^2*log(f)*log(x) - 2*b^2*log(x)*log(x^m))*log((e 
*x + d)^n)^2 - 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 + (b^2*e*m*n*x*log(x)^2 - 2*b^2*e*n*x*log(f)*log(x) + 2*b^2*d*log(c 
)*log(f) + 2*a*b*d*log(f) + 2*(b^2*e*log(c)*log(f) + a*b*e*log(f))*x - 2*( 
b^2*e*n*x*log(x) - b^2*d*log(c) - a*b*d - (b^2*e*log(c) + 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*l 
og(c)^2 + 2*a*b*e*log(c) + a^2*e)*x)*log(x^m))/(e*x^2 + d*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} \mathrm {log}\left (x^{m} f \right )}{x}d x \right ) b^{2} m +4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) \mathrm {log}\left (x^{m} f \right )}{x}d x \right ) a b m +\mathrm {log}\left (x^{m} f \right )^{2} a^{2}}{2 m} \] Input:

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

Output:

(2*int((log((d + e*x)**n*c)**2*log(x**m*f))/x,x)*b**2*m + 4*int((log((d + 
e*x)**n*c)*log(x**m*f))/x,x)*a*b*m + log(x**m*f)**2*a**2)/(2*m)