\(\int x^2 (a+b \log (c x^n)) (d+e \log (f (g+h x)^q)) \, dx\) [404]

Optimal result

Integrand size = 28, antiderivative size = 246 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=-\frac {a e g^2 q x}{3 h^2}+\frac {4 b e g^2 n q x}{9 h^2}-\frac {5 b e g n q x^2}{36 h}+\frac {2}{27} b e n q x^3-\frac {b e g^2 q x \log \left (c x^n\right )}{3 h^2}+\frac {e g q x^2 \left (a+b \log \left (c x^n\right )\right )}{6 h}-\frac {1}{9} e q x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e g^3 n q \log (g+h x)}{9 h^3}-\frac {1}{9} b n x^3 \left (d+e \log \left (f (g+h x)^q\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )+\frac {e g^3 q \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {h x}{g}\right )}{3 h^3}+\frac {b e g^3 n q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{3 h^3} \] Output:

-1/3*a*e*g^2*q*x/h^2+4/9*b*e*g^2*n*q*x/h^2-5/36*b*e*g*n*q*x^2/h+2/27*b*e*n 
*q*x^3-1/3*b*e*g^2*q*x*ln(c*x^n)/h^2+1/6*e*g*q*x^2*(a+b*ln(c*x^n))/h-1/9*e 
*q*x^3*(a+b*ln(c*x^n))-1/9*b*e*g^3*n*q*ln(h*x+g)/h^3-1/9*b*n*x^3*(d+e*ln(f 
*(h*x+g)^q))+1/3*x^3*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q))+1/3*e*g^3*q*(a+ 
b*ln(c*x^n))*ln(1+h*x/g)/h^3+1/3*b*e*g^3*n*q*polylog(2,-h*x/g)/h^3
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {-36 a e g^2 h q x+48 b e g^2 h n q x+18 a e g h^2 q x^2-15 b e g h^2 n q x^2+36 a d h^3 x^3-12 b d h^3 n x^3-12 a e h^3 q x^3+8 b e h^3 n q x^3+36 a e g^3 q \log (g+h x)-12 b e g^3 n q \log (g+h x)-36 b e g^3 n q \log (x) \log (g+h x)+36 a e h^3 x^3 \log \left (f (g+h x)^q\right )-12 b e h^3 n x^3 \log \left (f (g+h x)^q\right )+6 b \log \left (c x^n\right ) \left (6 d h^3 x^3+e h q x \left (-6 g^2+3 g h x-2 h^2 x^2\right )+6 e g^3 q \log (g+h x)+6 e h^3 x^3 \log \left (f (g+h x)^q\right )\right )+36 b e g^3 n q \log (x) \log \left (1+\frac {h x}{g}\right )+36 b e g^3 n q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{108 h^3} \] Input:

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

Output:

(-36*a*e*g^2*h*q*x + 48*b*e*g^2*h*n*q*x + 18*a*e*g*h^2*q*x^2 - 15*b*e*g*h^ 
2*n*q*x^2 + 36*a*d*h^3*x^3 - 12*b*d*h^3*n*x^3 - 12*a*e*h^3*q*x^3 + 8*b*e*h 
^3*n*q*x^3 + 36*a*e*g^3*q*Log[g + h*x] - 12*b*e*g^3*n*q*Log[g + h*x] - 36* 
b*e*g^3*n*q*Log[x]*Log[g + h*x] + 36*a*e*h^3*x^3*Log[f*(g + h*x)^q] - 12*b 
*e*h^3*n*x^3*Log[f*(g + h*x)^q] + 6*b*Log[c*x^n]*(6*d*h^3*x^3 + e*h*q*x*(- 
6*g^2 + 3*g*h*x - 2*h^2*x^2) + 6*e*g^3*q*Log[g + h*x] + 6*e*h^3*x^3*Log[f* 
(g + h*x)^q]) + 36*b*e*g^3*n*q*Log[x]*Log[1 + (h*x)/g] + 36*b*e*g^3*n*q*Po 
lyLog[2, -((h*x)/g)])/(108*h^3)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2889, 2793, 2009, 2842, 49, 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.

\(\displaystyle \int x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx\)

\(\Big \downarrow \) 2889

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{3} b n \int x^2 \left (d+e \log \left (f (g+h x)^q\right )\right )dx+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \left (-\frac {g^3 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^4}-\frac {g x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 h}+\frac {a g^2 x}{h^3}+\frac {b g^2 x \log \left (c x^n\right )}{h^3}-\frac {b g^3 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^4}-\frac {b g^2 n x}{h^3}+\frac {b g n x^2}{4 h^2}-\frac {b n x^3}{9 h}\right )\)

\(\Big \downarrow \) 2842

\(\displaystyle -\frac {1}{3} b n \left (\frac {1}{3} x^3 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \int \frac {x^3}{g+h x}dx\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \left (-\frac {g^3 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^4}-\frac {g x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 h}+\frac {a g^2 x}{h^3}+\frac {b g^2 x \log \left (c x^n\right )}{h^3}-\frac {b g^3 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^4}-\frac {b g^2 n x}{h^3}+\frac {b g n x^2}{4 h^2}-\frac {b n x^3}{9 h}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {1}{3} b n \left (\frac {1}{3} x^3 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \int \left (-\frac {g^3}{h^3 (g+h x)}+\frac {g^2}{h^3}-\frac {x g}{h^2}+\frac {x^2}{h}\right )dx\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \left (-\frac {g^3 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^4}-\frac {g x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 h}+\frac {a g^2 x}{h^3}+\frac {b g^2 x \log \left (c x^n\right )}{h^3}-\frac {b g^3 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^4}-\frac {b g^2 n x}{h^3}+\frac {b g n x^2}{4 h^2}-\frac {b n x^3}{9 h}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \left (-\frac {g^3 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^4}-\frac {g x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 h}+\frac {a g^2 x}{h^3}+\frac {b g^2 x \log \left (c x^n\right )}{h^3}-\frac {b g^3 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^4}-\frac {b g^2 n x}{h^3}+\frac {b g n x^2}{4 h^2}-\frac {b n x^3}{9 h}\right )-\frac {1}{3} b n \left (\frac {1}{3} x^3 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{3} e h q \left (-\frac {g^3 \log (g+h x)}{h^4}+\frac {g^2 x}{h^3}-\frac {g x^2}{2 h^2}+\frac {x^3}{3 h}\right )\right )\)

Input:

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

Output:

(x^3*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/3 - (b*n*(-1/3*(e*h*q* 
((g^2*x)/h^3 - (g*x^2)/(2*h^2) + x^3/(3*h) - (g^3*Log[g + h*x])/h^4)) + (x 
^3*(d + e*Log[f*(g + h*x)^q]))/3))/3 - (e*h*q*((a*g^2*x)/h^3 - (b*g^2*n*x) 
/h^3 + (b*g*n*x^2)/(4*h^2) - (b*n*x^3)/(9*h) + (b*g^2*x*Log[c*x^n])/h^3 - 
(g*x^2*(a + b*Log[c*x^n]))/(2*h^2) + (x^3*(a + b*Log[c*x^n]))/(3*h) - (g^3 
*(a + b*Log[c*x^n])*Log[1 + (h*x)/g])/h^4 - (b*g^3*n*PolyLog[2, -((h*x)/g) 
])/h^4))/3
 

Defintions of rubi rules used

rule 49

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

rule 2009

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

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], 
 (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, 
 f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer 
Q[r]))
 

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 = 155.59 (sec) , antiderivative size = 1112, normalized size of antiderivative = 4.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(1112\)

Input:

int(x^2*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q)),x,method=_RETURNVERBOSE)
 

Output:

-1/9*x^3*a*e*q-1/9*x^3*ln(c)*b*e*q-1/9*q*b*e*ln(x^n)*x^3+1/12*I*q/h*e*g*x^ 
2*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+1/12*I*q/h*e*g*x^2*b*Pi*csgn(I*x^n)*csgn( 
I*c*x^n)^2-1/6*I*q/h^2*e*g^2*x*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)-1/6*I*q/h^2* 
e*g^2*x*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I*q/h^3*e*g^3*ln(h*x+g)*b*Pi* 
csgn(I*c*x^n)^2*csgn(I*c)+1/6*I*q/h^3*e*g^3*ln(h*x+g)*b*Pi*csgn(I*x^n)*csg 
n(I*c*x^n)^2+49/108*b*e*g^3*n*q/h^3-1/3*n*b/h^3*e*g^3*q*ln(h*x+g)*ln(-h*x/ 
g)-1/18*I*q*e*x^3*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)-1/18*I*q*e*x^3*b*Pi*csgn( 
I*x^n)*csgn(I*c*x^n)^2+1/6*q/h*e*g*x^2*a+1/3*q/h^3*e*g^3*ln(h*x+g)*a-1/3*n 
*b/h^3*e*g^3*q*dilog(-h*x/g)+(1/3*b*e*x^3*ln(x^n)+1/18*e*x^3*(3*I*b*Pi*csg 
n(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3*I* 
b*Pi*csgn(I*c*x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)-2*n*b+6* 
a))*ln((h*x+g)^q)+(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 
)*(1/3*(I*Pi*b*csgn(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^3+2/3*b*x^3*ln(x^n)-2/9*b*n*x^3)+1/6*q/h*e*g*x^2*b*ln(c)-1/3*q/ 
h^2*e*g^2*x*b*ln(c)+1/3*q/h^3*e*g^3*ln(h*x+g)*b*ln(c)+1/18*I*q*e*x^3*b*Pi* 
csgn(I*c*x^n)^3-1/12*I*q/h*e*g*x^2*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c 
)+1/6*I*q/h^2*e*g^2*x*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/6*I*q/...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

integrate(x**2*(a+b*ln(c*x**n))*(d+e*ln(f*(h*x+g)**q)),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/9*b*d*n*x^3 + 1/3*a*e*x^3*log((h*x + g)^q*f) + 1/3*b*d*x^3*log(c*x^n) + 
 1/3*a*d*x^3 + 1/18*a*e*h*q*(6*g^3*log(h*x + g)/h^4 - (2*h^2*x^3 - 3*g*h*x 
^2 + 6*g^2*x)/h^3) - 1/18*b*e*((6*g^3*n*q*log(h*x + g)*log(x) - 2*(3*h^3*x 
^3*log(x^n) - (h^3*n - 3*h^3*log(c))*x^3)*log((h*x + g)^q) - (3*g*h^2*q*x^ 
2 - 6*g^2*h*q*x + 6*g^3*q*log(h*x + g) - 2*(h^3*q - 3*h^3*log(f))*x^3)*log 
(x^n))/h^3 - 18*integrate(1/18*(3*g^2*h*n*q*x + 6*g^3*n*q*log(x) + 6*g^3*n 
*q + 2*(2*h^3*n*q - 3*h^3*n*log(f) - 3*(h^3*q - 3*h^3*log(f))*log(c))*x^3 
- (g*h^2*n*q + 6*g*h^2*n*log(f) - 18*g*h^2*log(c)*log(f))*x^2)/(h^3*x + g* 
h^2), x))
 

Giac [F]

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

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

Output:

integrate((e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a)*x^2, x)
 

Mupad [F(-1)]

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

int(x^2*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)),x)
 

Output:

int(x^2*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)), x)
 

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {-36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{2}+g x}d x \right ) b e \,g^{4} n q +36 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b e \,h^{3} n \,x^{3}+36 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,g^{3} n +36 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,h^{3} n \,x^{3}-12 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,g^{3} n^{2}-12 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,h^{3} n^{2} x^{3}+18 \mathrm {log}\left (x^{n} c \right )^{2} b e \,g^{3} q +36 \,\mathrm {log}\left (x^{n} c \right ) b d \,h^{3} n \,x^{3}-36 \,\mathrm {log}\left (x^{n} c \right ) b e \,g^{2} h n q x +18 \,\mathrm {log}\left (x^{n} c \right ) b e g \,h^{2} n q \,x^{2}-12 \,\mathrm {log}\left (x^{n} c \right ) b e \,h^{3} n q \,x^{3}+36 a d \,h^{3} n \,x^{3}-36 a e \,g^{2} h n q x +18 a e g \,h^{2} n q \,x^{2}-12 a e \,h^{3} n q \,x^{3}-12 b d \,h^{3} n^{2} x^{3}+48 b e \,g^{2} h \,n^{2} q x -15 b e g \,h^{2} n^{2} q \,x^{2}+8 b e \,h^{3} n^{2} q \,x^{3}}{108 h^{3} n} \] Input:

int(x^2*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x)
 

Output:

( - 36*int(log(x**n*c)/(g*x + h*x**2),x)*b*e*g**4*n*q + 36*log((g + h*x)** 
q*f)*log(x**n*c)*b*e*h**3*n*x**3 + 36*log((g + h*x)**q*f)*a*e*g**3*n + 36* 
log((g + h*x)**q*f)*a*e*h**3*n*x**3 - 12*log((g + h*x)**q*f)*b*e*g**3*n**2 
 - 12*log((g + h*x)**q*f)*b*e*h**3*n**2*x**3 + 18*log(x**n*c)**2*b*e*g**3* 
q + 36*log(x**n*c)*b*d*h**3*n*x**3 - 36*log(x**n*c)*b*e*g**2*h*n*q*x + 18* 
log(x**n*c)*b*e*g*h**2*n*q*x**2 - 12*log(x**n*c)*b*e*h**3*n*q*x**3 + 36*a* 
d*h**3*n*x**3 - 36*a*e*g**2*h*n*q*x + 18*a*e*g*h**2*n*q*x**2 - 12*a*e*h**3 
*n*q*x**3 - 12*b*d*h**3*n**2*x**3 + 48*b*e*g**2*h*n**2*q*x - 15*b*e*g*h**2 
*n**2*q*x**2 + 8*b*e*h**3*n**2*q*x**3)/(108*h**3*n)