3.4.73 \(\int \log (f x^m) (a+b \log (c (d+e x)^n))^3 \, dx\) [373]

3.4.73.1 Optimal result

Integrand size = 23, antiderivative size = 522 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )}{e} \]

output

-12*a*b^2*m*n^2*x+18*b^3*m*n^3*x-6*b^2*m*n^2*(-b*n+a)*x+6*a*b^2*n^2*x*ln(f 
*x^m)-6*b^3*n^3*x*ln(f*x^m)-18*b^3*m*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e-6*b^3*d 
*m*n^2*ln(-e*x/d)*ln(c*(e*x+d)^n)/e+6*b^3*n^2*(e*x+d)*ln(f*x^m)*ln(c*(e*x+ 
d)^n)/e+6*b*m*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e+3*b*d*m*n*ln(-e*x/d)*(a+ 
b*ln(c*(e*x+d)^n))^2/e-3*b*n*(e*x+d)*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/e-m 
*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e-d*m*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))^3/ 
e+(e*x+d)*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^3/e-6*b^3*d*m*n^3*polylog(2,1+e* 
x/d)/e+6*b^2*d*m*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(2,1+e*x/d)/e-3*b*d*m*n* 
(a+b*ln(c*(e*x+d)^n))^2*polylog(2,1+e*x/d)/e-6*b^3*d*m*n^3*polylog(3,1+e*x 
/d)/e+6*b^2*d*m*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,1+e*x/d)/e-6*b^3*d*m*n 
^3*polylog(4,1+e*x/d)/e
 

3.4.73.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1163\) vs. \(2(522)=1044\).

Time = 0.46 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.23 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx =\text {Too large to display} \]

input

Integrate[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3,x]
 

output

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

3.4.73.3 Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2870, 2009}

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.

\(\displaystyle \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx\)

\(\Big \downarrow \) 2870

\(\displaystyle -m \int \left (\frac {6 n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e x}+6 n^2 (a-b n) b^2-\frac {3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e x}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e x}\right )dx+6 a b^2 n^2 x \log \left (f x^m\right )-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 6 a b^2 n^2 x \log \left (f x^m\right )-m \left (-\frac {6 b^2 d n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {6 b^2 d n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+12 a b^2 n^2 x+6 b^2 n^2 x (a-b n)+\frac {3 b d n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {6 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b d n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {d \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {18 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 d n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 d n^3 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}+\frac {6 b^3 d n^3 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e}+\frac {6 b^3 d n^3 \operatorname {PolyLog}\left (4,\frac {e x}{d}+1\right )}{e}-18 b^3 n^3 x\right )-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )\)

input

Int[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3,x]
 

output

6*a*b^2*n^2*x*Log[f*x^m] - 6*b^3*n^3*x*Log[f*x^m] + (6*b^3*n^2*(d + e*x)*L 
og[f*x^m]*Log[c*(d + e*x)^n])/e - (3*b*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c 
*(d + e*x)^n])^2)/e + ((d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3)/ 
e - m*(12*a*b^2*n^2*x - 18*b^3*n^3*x + 6*b^2*n^2*(a - b*n)*x + (18*b^3*n^2 
*(d + e*x)*Log[c*(d + e*x)^n])/e + (6*b^3*d*n^2*Log[-((e*x)/d)]*Log[c*(d + 
 e*x)^n])/e - (6*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e - (3*b*d*n* 
Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/e + ((d + e*x)*(a + b*Log[c* 
(d + e*x)^n])^3)/e + (d*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^3)/e + 
(6*b^3*d*n^3*PolyLog[2, 1 + (e*x)/d])/e - (6*b^2*d*n^2*(a + b*Log[c*(d + e 
*x)^n])*PolyLog[2, 1 + (e*x)/d])/e + (3*b*d*n*(a + b*Log[c*(d + e*x)^n])^2 
*PolyLog[2, 1 + (e*x)/d])/e + (6*b^3*d*n^3*PolyLog[3, 1 + (e*x)/d])/e - (6 
*b^2*d*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, 1 + (e*x)/d])/e + (6*b^3* 
d*n^3*PolyLog[4, 1 + (e*x)/d])/e)
 

3.4.73.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2870

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

3.4.73.4 Maple [F]

\[\int \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}d x\]

input

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^3,x)
 

output

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^3,x)
 

3.4.73.5 Fricas [F]

\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")
 

output

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

3.4.73.6 Sympy [F(-1)]

Timed out. \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Timed out} \]

input

integrate(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**3,x)
 

output

Timed out
 

3.4.73.7 Maxima [F]

\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")
 

output

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

3.4.73.8 Giac [F]

\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]

input

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")
 

output

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

3.4.73.9 Mupad [F(-1)]

Timed out. \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int \ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3 \,d x \]

input

int(log(f*x^m)*(a + b*log(c*(d + e*x)^n))^3,x)
 

output

int(log(f*x^m)*(a + b*log(c*(d + e*x)^n))^3, x)