\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x^4} \, dx\) [385]

Optimal result

Integrand size = 32, antiderivative size = 234 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {b e^2 g n^2}{3 d^2 x}-\frac {b e^3 g n^2 \log (x)}{d^3}+\frac {b e^3 g n^2 \log (d+e x)}{3 d^3}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}-\frac {e n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {e^2 n (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {e^3 n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3}-\frac {2 b e^3 g n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {a f}{3 x^3}+\frac {1}{3} b e f n \left (-\frac {1}{2 d x^2}+\frac {e}{d^2 x}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log (d+e x)}{d^3}\right )+\frac {1}{3} a e g n \left (-\frac {1}{2 d x^2}+\frac {e}{d^2 x}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log (d+e x)}{d^3}\right )-\frac {b f \log \left (c (d+e x)^n\right )}{3 x^3}-\frac {a g \log \left (c (d+e x)^n\right )}{3 x^3}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{3 x^3}+\frac {2}{3} b e g n \left (-\frac {e n}{2 d^2 x}-\frac {3 e^2 n \log (x)}{2 d^3}+\frac {3 e^2 n \log (d+e x)}{2 d^3}-\frac {\log \left (c (d+e x)^n\right )}{2 d x^2}+\frac {e \log \left (c (d+e x)^n\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d^3}-\frac {e^2 \log ^2\left (c (d+e x)^n\right )}{2 d^3 n}+\frac {e^2 n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d^3}\right ) \] Input:

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2883, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

\(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{x^4} \, dx\)

\(\Big \downarrow \) 2883

\(\displaystyle \frac {1}{3} e n \int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{x^3 (d+e x)}dx-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {1}{3} n \int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{x^3 (d+e x)}d(d+e x)-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} n \int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{x^3 (d+e x)}d(d+e x)-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} e^3 n \int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^3 x^3 (d+e x)}d(d+e x)-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2789

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^3 x^3}d(d+e x)}{d}+\frac {\int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^2 x^2 (d+e x)}d(d+e x)}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2756

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \int \frac {1}{e^2 x^2 (d+e x)}d(d+e x)}{d}+\frac {\int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^2 x^2 (d+e x)}d(d+e x)}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 54

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \int \left (\frac {1}{e^2 x^2 d}-\frac {1}{e x d^2}+\frac {1}{(d+e x) d^2}\right )d(d+e x)}{d}+\frac {\int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^2 x^2 (d+e x)}d(d+e x)}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^2 x^2 (d+e x)}d(d+e x)}{d}+\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \left (-\frac {\log (-e x)}{d^2}+\frac {\log (d+e x)}{d^2}-\frac {1}{d e x}\right )}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2789

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {\int \frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e^2 x^2}d(d+e x)}{d}+\frac {\int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e x (d+e x)}d(d+e x)}{d}}{d}+\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \left (-\frac {\log (-e x)}{d^2}+\frac {\log (d+e x)}{d^2}-\frac {1}{d e x}\right )}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2751

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {-\frac {2 b g n \int -\frac {1}{e x}d(d+e x)}{d}-\frac {(d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{d e x}}{d}+\frac {\int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e x (d+e x)}d(d+e x)}{d}}{d}+\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \left (-\frac {\log (-e x)}{d^2}+\frac {\log (d+e x)}{d^2}-\frac {1}{d e x}\right )}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {\int -\frac {b f+a g+2 b g \log \left (c (d+e x)^n\right )}{e x (d+e x)}d(d+e x)}{d}+\frac {\frac {2 b g n \log (-e x)}{d}-\frac {(d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{d e x}}{d}}{d}+\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \left (-\frac {\log (-e x)}{d^2}+\frac {\log (d+e x)}{d^2}-\frac {1}{d e x}\right )}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2779

\(\displaystyle -\frac {1}{3} e^3 n \left (\frac {\frac {\frac {2 b g n \int \frac {\log \left (1-\frac {d}{d+e x}\right )}{d+e x}d(d+e x)}{d}-\frac {\log \left (1-\frac {d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{d}}{d}+\frac {\frac {2 b g n \log (-e x)}{d}-\frac {(d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{d e x}}{d}}{d}+\frac {\frac {a g+2 b g \log \left (c (d+e x)^n\right )+b f}{2 e^2 x^2}-b g n \left (-\frac {\log (-e x)}{d^2}+\frac {\log (d+e x)}{d^2}-\frac {1}{d e x}\right )}{d}\right )-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}\)

\(\Big \downarrow \) 2838

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

Input:

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

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

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2883

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.14 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.74

method	result	size
risch	\(\left (i \pi b g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i \pi b g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right ) g +a g +b f \right ) \left (-\frac {\ln \left (\left (e x +d \right )^{n}\right )}{3 x^{3}}+\frac {e n \left (-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )}{3}\right )-\frac {\ln \left (\left (e x +d \right )^{n}\right )^{2} b g}{3 x^{3}}-\frac {2 b g \,e^{3} n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{3 d^{3}}-\frac {b g e n \ln \left (\left (e x +d \right )^{n}\right )}{3 d \,x^{2}}+\frac {2 b g \,e^{3} n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )}{3 d^{3}}+\frac {2 b g \,e^{2} n \ln \left (\left (e x +d \right )^{n}\right )}{3 d^{2} x}+\frac {b \,e^{3} g \,n^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {b \,e^{2} g \,n^{2}}{3 d^{2} x}-\frac {b \,e^{3} g \,n^{2} \ln \left (x \right )}{d^{3}}-\frac {2 b g \,e^{3} n^{2} \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{3 d^{3}}-\frac {2 b g \,e^{3} n^{2} \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{3 d^{3}}+\frac {b g \,e^{3} n^{2} \ln \left (e x +d \right )^{2}}{3 d^{3}}-\frac {\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) \left (i g \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i g \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i g \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i g \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 g \ln \left (c \right )+2 f \right )}{12 x^{3}}\)	\(642\)

Input:

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))/x^4,x,method=_RETURNVERBOS 
E)
 

Output:

(I*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*g*csgn(I*(e*x+d)^ 
n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*Pi*b*g*csgn(I*c*(e*x+d)^n)^3+I*Pi*b*g*c 
sgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)*g+a*g+b*f)*(-1/3*ln((e*x+d)^n)/x^ 
3+1/3*e*n*(-e^2/d^3*ln(e*x+d)-1/2/d/x^2+e^2/d^3*ln(x)+e/d^2/x))-1/3/x^3*ln 
((e*x+d)^n)^2*b*g-2/3*b*g*e^3*n*ln((e*x+d)^n)/d^3*ln(e*x+d)-1/3*b*g*e*n*ln 
((e*x+d)^n)/d/x^2+2/3*b*g*e^3*n*ln((e*x+d)^n)/d^3*ln(x)+2/3*b*g*e^2*n*ln(( 
e*x+d)^n)/d^2/x+b*e^3*g*n^2*ln(e*x+d)/d^3-1/3*b*e^2*g*n^2/d^2/x-b*e^3*g*n^ 
2*ln(x)/d^3-2/3*b*g*e^3*n^2/d^3*dilog((e*x+d)/d)-2/3*b*g*e^3*n^2/d^3*ln(x) 
*ln((e*x+d)/d)+1/3*b*g*e^3*n^2/d^3*ln(e*x+d)^2-1/12*(I*b*Pi*csgn(I*(e*x+d) 
^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csg 
n(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c) 
+2*b*ln(c)+2*a)*(I*g*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*g*Pi*csg 
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*g*Pi*csgn(I*c*(e*x+d)^n)^3+ 
I*g*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*g*ln(c)+2*f)/x^3
 

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm=" 
fricas")
 

Output:

integral((b*g*log((e*x + d)^n*c)^2 + a*f + (b*f + a*g)*log((e*x + d)^n*c)) 
/x^4, x)
 

Sympy [F]

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

integrate((a+b*ln(c*(e*x+d)**n))*(f+g*ln(c*(e*x+d)**n))/x**4,x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))*(f + g*log(c*(d + e*x)**n))/x**4, x)
 

Maxima [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm=" 
maxima")
 

Output:

-1/6*b*e*f*n*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - d)/(d^2 
*x^2)) - 1/6*a*e*g*n*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - 
 d)/(d^2*x^2)) - 1/3*b*g*(log((e*x + d)^n)^2/x^3 - 3*integrate(1/3*(3*e*x* 
log(c)^2 + 3*d*log(c)^2 + 2*((e*n + 3*e*log(c))*x + 3*d*log(c))*log((e*x + 
 d)^n))/(e*x^5 + d*x^4), x)) - 1/3*b*f*log((e*x + d)^n*c)/x^3 - 1/3*a*g*lo 
g((e*x + d)^n*c)/x^3 - 1/3*a*f/x^3
 

Giac [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm=" 
giac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)*(g*log((e*x + d)^n*c) + f)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e \,x^{2}+d x}d x \right ) b d \,e^{3} g n \,x^{3}-2 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b \,d^{3} g -2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a \,d^{3} g -2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a \,e^{3} g \,x^{3}-2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{3} f -2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{2} e g n x +4 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b d \,e^{2} g n \,x^{2}-2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{3} f \,x^{3}+6 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{3} g n \,x^{3}+2 \,\mathrm {log}\left (x \right ) a \,e^{3} g n \,x^{3}+2 \,\mathrm {log}\left (x \right ) b \,e^{3} f n \,x^{3}-6 \,\mathrm {log}\left (x \right ) b \,e^{3} g \,n^{2} x^{3}-2 a \,d^{3} f -a \,d^{2} e g n x +2 a d \,e^{2} g n \,x^{2}-b \,d^{2} e f n x +2 b d \,e^{2} f n \,x^{2}-2 b d \,e^{2} g \,n^{2} x^{2}}{6 d^{3} x^{3}} \] Input:

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

Output:

(4*int(log((d + e*x)**n*c)/(d*x + e*x**2),x)*b*d*e**3*g*n*x**3 - 2*log((d 
+ e*x)**n*c)**2*b*d**3*g - 2*log((d + e*x)**n*c)*a*d**3*g - 2*log((d + e*x 
)**n*c)*a*e**3*g*x**3 - 2*log((d + e*x)**n*c)*b*d**3*f - 2*log((d + e*x)** 
n*c)*b*d**2*e*g*n*x + 4*log((d + e*x)**n*c)*b*d*e**2*g*n*x**2 - 2*log((d + 
 e*x)**n*c)*b*e**3*f*x**3 + 6*log((d + e*x)**n*c)*b*e**3*g*n*x**3 + 2*log( 
x)*a*e**3*g*n*x**3 + 2*log(x)*b*e**3*f*n*x**3 - 6*log(x)*b*e**3*g*n**2*x** 
3 - 2*a*d**3*f - a*d**2*e*g*n*x + 2*a*d*e**2*g*n*x**2 - b*d**2*e*f*n*x + 2 
*b*d*e**2*f*n*x**2 - 2*b*d*e**2*g*n**2*x**2)/(6*d**3*x**3)