\(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))^2}{x} \, dx\) [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=\frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-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+2 b n \left (-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 ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\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 (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (3,\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (4,\frac {e x}{d+e x}\right )-\operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )\right )\right ) \]

[Out]

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*p
olylog(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*ln(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*(ln(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)+2
4*(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)-24*polylog(4,-e*x/d)-24*pol
ylog(4,e*x/(e*x+d))+24*polylog(4,1+e*x/d))

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (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=\frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-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+2 b n \left (-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 ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\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 (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (3,\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (4,\frac {e x}{d+e x}\right )-\operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )\right )\right ) \]

[In]

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

[Out]

(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*Lo
g[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)] + PolyLo
g[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))] + Lo
g[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((e*x)/(d + e*x))]*PolyLog[3, (e*x)/(d + e*x)] + 24*(-Log[x] + Lo
g[-((e*x)/(d + e*x))])*PolyLog[3, 1 + (e*x)/d] - 24*(PolyLog[4, -((e*x)/d)] + PolyLog[4, (e*x)/(d + e*x)] - Po
lyLog[4, 1 + (e*x)/d])))/12

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-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*l
og(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)*lo
g(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^2 + d*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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