\(\int \frac {(a+b \log (c x^n)) (d+e \log (f (g+h x)^q))}{x^3} \, dx\) [409]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^3} \, dx=-\frac {3 b e h n q}{4 g x}-\frac {b e h^2 n q \log (x)}{4 g^2}-\frac {e h q \left (a+b \log \left (c x^n\right )\right )}{2 g x}+\frac {e h^2 q \log \left (1+\frac {g}{h x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 g^2}+\frac {b e h^2 n q \log (g+h x)}{4 g^2}-\frac {b n \left (d+e \log \left (f (g+h x)^q\right )\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{2 x^2}-\frac {b e h^2 n q \operatorname {PolyLog}\left (2,-\frac {g}{h x}\right )}{2 g^2} \] Output:

-3/4*b*e*h*n*q/g/x-1/4*b*e*h^2*n*q*ln(x)/g^2-1/2*e*h*q*(a+b*ln(c*x^n))/g/x 
+1/2*e*h^2*q*ln(1+g/h/x)*(a+b*ln(c*x^n))/g^2+1/4*b*e*h^2*n*q*ln(h*x+g)/g^2 
-1/4*b*n*(d+e*ln(f*(h*x+g)^q))/x^2-1/2*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q 
))/x^2-1/2*b*e*h^2*n*q*polylog(2,-g/h/x)/g^2
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^3} \, dx=-\frac {2 a d g^2+b d g^2 n+2 a e g h q x+3 b e g h n q x-b e h^2 n q x^2 \log ^2(x)+2 b d g^2 \log \left (c x^n\right )+2 b e g h q x \log \left (c x^n\right )-2 a e h^2 q x^2 \log (g+h x)-b e h^2 n q x^2 \log (g+h x)-2 b e h^2 q x^2 \log \left (c x^n\right ) \log (g+h x)+2 a e g^2 \log \left (f (g+h x)^q\right )+b e g^2 n \log \left (f (g+h x)^q\right )+2 b e g^2 \log \left (c x^n\right ) \log \left (f (g+h x)^q\right )+e h^2 q x^2 \log (x) \left (2 a+b n+2 b \log \left (c x^n\right )+2 b n \log (g+h x)-2 b n \log \left (1+\frac {h x}{g}\right )\right )-2 b e h^2 n q x^2 \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{4 g^2 x^2} \] Input:

Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2889, 2780, 2741, 2779, 2838, 2842, 54, 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.

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

\(\Big \downarrow \) 2889

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

\(\Big \downarrow \) 2780

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

\(\Big \downarrow \) 2741

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 2842

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/x^3,x]
 

Output:

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

Defintions of rubi rules used

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 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -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 2780

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

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 2842

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ 
))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( 
g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1)))   Int[(f + g*x)^(q + 1)/(d + e*x 
), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && 
NeQ[q, -1]
 

rule 2889

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^( 
r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1)), x 
] + (-Simp[g*j*(m/(r + 1))   Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i 
 + j*x)), x], x] - Simp[b*e*n*(p/(r + 1))   Int[x^(r + 1)*(a + b*Log[c*(d + 
 e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[ 
{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (E 
qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 33.98 (sec) , antiderivative size = 988, normalized size of antiderivative = 5.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(988\)

Input:

int((a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q))/x^3,x,method=_RETURNVERBOSE)
 

Output:

(1/4*I*e*Pi*csgn(I*(h*x+g)^q)*csgn(I*f*(h*x+g)^q)^2-1/4*I*e*Pi*csgn(I*(h*x 
+g)^q)*csgn(I*f*(h*x+g)^q)*csgn(I*f)-1/4*I*e*Pi*csgn(I*f*(h*x+g)^q)^3+1/4* 
I*e*Pi*csgn(I*f*(h*x+g)^q)^2*csgn(I*f)+1/2*ln(f)*e+1/2*d)*(-1/2*(I*Pi*b*cs 
gn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi* 
b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)/x^2-b/x^ 
2*ln(x^n)-1/2*b*n/x^2)+(-1/2*b*e/x^2*ln(x^n)-1/4*(I*Pi*b*e*csgn(I*x^n)*csg 
n(I*c*x^n)^2-I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*e*csgn(I* 
c*x^n)^3+I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)*b*e+b*e*n+2*e*a)/x^2)* 
ln((h*x+g)^q)-1/4*I*h^2*q*e/g^2*ln(h*x+g)*Pi*csgn(I*c*x^n)^3*b+1/4*I*h*q*e 
/g/x*Pi*csgn(I*c*x^n)^3*b+1/4*I*h^2*q*e/g^2*ln(x)*Pi*csgn(I*c*x^n)^3*b-1/2 
*h^2*q*b*e*n/g^2*ln(h*x+g)*ln(-h*x/g)+1/2*h^2*q*e/g^2*ln(h*x+g)*a-1/2*h^2* 
q*e/g^2*ln(x)*a-1/4*I*h*q*e/g/x*Pi*csgn(I*c*x^n)^2*csgn(I*c)*b-1/4*I*h*q*e 
/g/x*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*b+1/4*I*h^2*q*e/g^2*ln(h*x+g)*Pi*csgn( 
I*x^n)*csgn(I*c*x^n)^2*b+1/4*I*h^2*q*e/g^2*ln(h*x+g)*Pi*csgn(I*c*x^n)^2*cs 
gn(I*c)*b-1/4*I*h^2*q*e/g^2*ln(x)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*b-1/4*I*h 
^2*q*e/g^2*ln(x)*Pi*csgn(I*c*x^n)^2*csgn(I*c)*b-1/4*I*h^2*q*e/g^2*ln(h*x+g 
)*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*b+1/4*I*h^2*q*e/g^2*ln(x)*Pi*csgn 
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*b+1/4*I*h*q*e/g/x*Pi*csgn(I*x^n)*csgn(I*c* 
x^n)*csgn(I*c)*b+1/2*h^2*q*b*e*ln(x^n)/g^2*ln(h*x+g)-1/2*h^2*q*b*e*ln(x^n) 
/g^2*ln(x)-1/2*h^2*q*b*e*n/g^2*dilog(-h*x/g)+1/4*h^2*q*b*e*n/g^2*ln(x)^...
 

Fricas [F]

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

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

Output:

integral((b*d*log(c*x^n) + a*d + (b*e*log(c*x^n) + a*e)*log((h*x + g)^q*f) 
)/x^3, x)
 

Sympy [F(-1)]

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

integrate((a+b*ln(c*x**n))*(d+e*ln(f*(h*x+g)**q))/x**3,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate((e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a)/x^3, x)
 

Mupad [F(-1)]

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

int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)))/x^3,x)
                                                                                    
                                                                                    
 

Output:

int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)))/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^3} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{4}+g \,x^{3}}d x \right ) b e \,g^{3} q \,x^{2}-4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b e \,g^{2}-4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,g^{2}+4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,h^{2} x^{2}-2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,g^{2} n +2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,h^{2} n \,x^{2}-4 \,\mathrm {log}\left (x^{n} c \right ) b d \,g^{2}-2 \,\mathrm {log}\left (x^{n} c \right ) b e \,g^{2} q -4 \,\mathrm {log}\left (x \right ) a e \,h^{2} q \,x^{2}-2 \,\mathrm {log}\left (x \right ) b e \,h^{2} n q \,x^{2}-4 a d \,g^{2}-4 a e g h q x -2 b d \,g^{2} n -b e \,g^{2} n q -2 b e g h n q x}{8 g^{2} x^{2}} \] Input:

int((a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q))/x^3,x)
 

Output:

( - 4*int(log(x**n*c)/(g*x**3 + h*x**4),x)*b*e*g**3*q*x**2 - 4*log((g + h* 
x)**q*f)*log(x**n*c)*b*e*g**2 - 4*log((g + h*x)**q*f)*a*e*g**2 + 4*log((g 
+ h*x)**q*f)*a*e*h**2*x**2 - 2*log((g + h*x)**q*f)*b*e*g**2*n + 2*log((g + 
 h*x)**q*f)*b*e*h**2*n*x**2 - 4*log(x**n*c)*b*d*g**2 - 2*log(x**n*c)*b*e*g 
**2*q - 4*log(x)*a*e*h**2*q*x**2 - 2*log(x)*b*e*h**2*n*q*x**2 - 4*a*d*g**2 
 - 4*a*e*g*h*q*x - 2*b*d*g**2*n - b*e*g**2*n*q - 2*b*e*g*h*n*q*x)/(8*g**2* 
x**2)