\(\int \frac {a+b \log (c x^n)}{x (d+e x)^4} \, dx\) [59]

Optimal result

Integrand size = 21, antiderivative size = 174 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=-\frac {b n}{6 d^2 (d+e x)^2}-\frac {5 b n}{6 d^3 (d+e x)}-\frac {5 b n \log (x)}{6 d^4}+\frac {a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {11 b n \log (d+e x)}{6 d^4}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4} \] Output:

-1/6*b*n/d^2/(e*x+d)^2-5/6*b*n/d^3/(e*x+d)-5/6*b*n*ln(x)/d^4+1/3*(a+b*ln(c 
*x^n))/d/(e*x+d)^3+1/2*(a+b*ln(c*x^n))/d^2/(e*x+d)^2-e*x*(a+b*ln(c*x^n))/d 
^4/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))/d^4+11/6*b*n*ln(e*x+d)/d^4+b*n*poly 
log(2,-d/e/x)/d^4
 

Mathematica [F(-1)]

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

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

Output:

$Aborted
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2789, 2756, 54, 2009, 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.

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

\(\Big \downarrow \) 2789

\(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4}dx}{d}\)

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.17 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.82

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{4}}+\frac {b \ln \left (x^{n}\right )}{d^{3} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right )}{2 d^{2} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right )}{3 d \left (e x +d \right )^{3}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{4}}-\frac {5 b n}{6 d^{3} \left (e x +d \right )}-\frac {b n}{6 d^{2} \left (e x +d \right )^{2}}+\frac {11 b n \ln \left (e x +d \right )}{6 d^{4}}-\frac {11 b n \ln \left (x \right )}{6 d^{4}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{4}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{4}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{4}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{4}}+\frac {1}{d^{3} \left (e x +d \right )}+\frac {1}{2 d^{2} \left (e x +d \right )^{2}}+\frac {1}{3 d \left (e x +d \right )^{3}}+\frac {\ln \left (x \right )}{d^{4}}\right )\)	\(316\)

Input:

int((a+b*ln(c*x^n))/x/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

-b*ln(x^n)/d^4*ln(e*x+d)+b*ln(x^n)/d^3/(e*x+d)+1/2*b*ln(x^n)/d^2/(e*x+d)^2 
+1/3*b*ln(x^n)/d/(e*x+d)^3+b*ln(x^n)/d^4*ln(x)-5/6*b*n/d^3/(e*x+d)-1/6*b*n 
/d^2/(e*x+d)^2+11/6*b*n*ln(e*x+d)/d^4-11/6*b*n*ln(x)/d^4-1/2*b*n/d^4*ln(x) 
^2+b*n/d^4*ln(e*x+d)*ln(-e*x/d)+b*n/d^4*dilog(-e*x/d)+(1/2*I*Pi*b*csgn(I*x 
^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*P 
i*b*csgn(I*c*x^n)^3+1/2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*(-1/d^ 
4*ln(e*x+d)+1/d^3/(e*x+d)+1/2/d^2/(e*x+d)^2+1/3/d/(e*x+d)^3+1/d^4*ln(x))
 

Fricas [F]

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

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

Output:

integral((b*log(c*x^n) + a)/(e^4*x^5 + 4*d*e^3*x^4 + 6*d^2*e^2*x^3 + 4*d^3 
*e*x^2 + d^4*x), x)
 

Sympy [A] (verification not implemented)

Time = 66.28 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.93 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-a*e*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d - a*e* 
Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/d**2 - a*e*Pi 
ecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**3 - a*e*Piecewis 
e((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**4 + a*log(x)/d**4 - b*e**3*n 
*Piecewise((-1/(e**4*x), Eq(d, 0)), (-3*d/(6*d**2*e**3 + 12*d*e**4*x + 6*e 
**5*x**2) - 4*e*x/(6*d**2*e**3 + 12*d*e**4*x + 6*e**5*x**2) - log(d + e*x) 
/(3*d*e**3), True))/d**3 + b*e**3*Piecewise((1/(e**4*x), Eq(d, 0)), (-1/(3 
*d*(d/x + e)**3), True))*log(c*x**n)/d**3 + 3*b*e**2*n*Piecewise((-1/(e**3 
*x), Eq(d, 0)), (-1/(2*d*e**2 + 2*e**3*x) - log(d + e*x)/(2*d*e**2), True) 
)/d**3 - 3*b*e**2*Piecewise((1/(e**3*x), Eq(d, 0)), (-1/(2*d*(d/x + e)**2) 
, True))*log(c*x**n)/d**3 - 3*b*e*n*Piecewise((-1/(e**2*x), Eq(d, 0)), (-l 
og(d**2 + d*e*x)/(d*e), True))/d**3 + 3*b*e*Piecewise((1/(e**2*x), Eq(d, 0 
)), (-1/(d**2/x + d*e), True))*log(c*x**n)/d**3 + b*n*Piecewise((-1/(e*x), 
 Eq(d, 0)), (Piecewise((polylog(2, d*exp_polar(I*pi)/(e*x)), (Abs(x) < 1) 
& (1/Abs(x) < 1)), (log(e)*log(x) + polylog(2, d*exp_polar(I*pi)/(e*x)), A 
bs(x) < 1), (-log(e)*log(1/x) + polylog(2, d*exp_polar(I*pi)/(e*x)), 1/Abs 
(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1 
), ()), ((), (0, 0)), x)*log(e) + polylog(2, d*exp_polar(I*pi)/(e*x)), Tru 
e))/d, True))/d**3 - b*Piecewise((1/(e*x), Eq(d, 0)), (log(d/x + e)/d, Tru 
e))*log(c*x**n)/d**3
 

Maxima [F]

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

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

Output:

1/6*a*((6*e^2*x^2 + 15*d*e*x + 11*d^2)/(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3*d^ 
5*e*x + d^6) - 6*log(e*x + d)/d^4 + 6*log(x)/d^4) + b*integrate((log(c) + 
log(x^n))/(e^4*x^5 + 4*d*e^3*x^4 + 6*d^2*e^2*x^3 + 4*d^3*e*x^2 + d^4*x), x 
)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\frac {6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{7}+18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{6} e x +18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{5} e^{2} x^{2}+6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{4} e^{3} x^{3}-6 \,\mathrm {log}\left (e x +d \right ) a \,d^{3}-18 \,\mathrm {log}\left (e x +d \right ) a \,d^{2} e x -18 \,\mathrm {log}\left (e x +d \right ) a d \,e^{2} x^{2}-6 \,\mathrm {log}\left (e x +d \right ) a \,e^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) a \,d^{3}+18 \,\mathrm {log}\left (x \right ) a \,d^{2} e x +18 \,\mathrm {log}\left (x \right ) a d \,e^{2} x^{2}+6 \,\mathrm {log}\left (x \right ) a \,e^{3} x^{3}+9 a \,d^{3}+9 a \,d^{2} e x -2 a \,e^{3} x^{3}}{6 d^{4} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

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

Output:

(6*int(log(x**n*c)/(d**4*x + 4*d**3*e*x**2 + 6*d**2*e**2*x**3 + 4*d*e**3*x 
**4 + e**4*x**5),x)*b*d**7 + 18*int(log(x**n*c)/(d**4*x + 4*d**3*e*x**2 + 
6*d**2*e**2*x**3 + 4*d*e**3*x**4 + e**4*x**5),x)*b*d**6*e*x + 18*int(log(x 
**n*c)/(d**4*x + 4*d**3*e*x**2 + 6*d**2*e**2*x**3 + 4*d*e**3*x**4 + e**4*x 
**5),x)*b*d**5*e**2*x**2 + 6*int(log(x**n*c)/(d**4*x + 4*d**3*e*x**2 + 6*d 
**2*e**2*x**3 + 4*d*e**3*x**4 + e**4*x**5),x)*b*d**4*e**3*x**3 - 6*log(d + 
 e*x)*a*d**3 - 18*log(d + e*x)*a*d**2*e*x - 18*log(d + e*x)*a*d*e**2*x**2 
- 6*log(d + e*x)*a*e**3*x**3 + 6*log(x)*a*d**3 + 18*log(x)*a*d**2*e*x + 18 
*log(x)*a*d*e**2*x**2 + 6*log(x)*a*e**3*x**3 + 9*a*d**3 + 9*a*d**2*e*x - 2 
*a*e**3*x**3)/(6*d**4*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))