\(\int x^2 (a+b \log (c x^n))^2 (d+e \log (f (g+h x)^q)) \, dx\) [410]

Optimal result

Integrand size = 30, antiderivative size = 525 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {8 a b e g^2 n q x}{9 h^2}-\frac {26 b^2 e g^2 n^2 q x}{27 h^2}+\frac {19 b^2 e g n^2 q x^2}{108 h}+\frac {2}{27} b^2 d n^2 x^3-\frac {2}{27} b^2 e n^2 q x^3+\frac {8 b^2 e g^2 n q x \log \left (c x^n\right )}{9 h^2}-\frac {5 b e g n q x^2 \left (a+b \log \left (c x^n\right )\right )}{18 h}-\frac {2}{9} b d n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4}{27} b e n q x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e g^2 q x \left (a+b \log \left (c x^n\right )\right )^2}{3 h^2}+\frac {e g q x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 h}+\frac {1}{3} d x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{9} e q x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e g^3 n^2 q \log (g+h x)}{27 h^3}+\frac {2}{27} b^2 e n^2 x^3 \log \left (f (g+h x)^q\right )-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (f (g+h x)^q\right )+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (f (g+h x)^q\right )-\frac {2 b e g^3 n q \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {h x}{g}\right )}{9 h^3}+\frac {e g^3 q \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {h x}{g}\right )}{3 h^3}-\frac {2 b^2 e g^3 n^2 q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{9 h^3}+\frac {2 b e g^3 n q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{3 h^3}-\frac {2 b^2 e g^3 n^2 q \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right )}{3 h^3} \] Output:

8/9*a*b*e*g^2*n*q*x/h^2-26/27*b^2*e*g^2*n^2*q*x/h^2+19/108*b^2*e*g*n^2*q*x 
^2/h+2/27*b^2*d*n^2*x^3-2/27*b^2*e*n^2*q*x^3+8/9*b^2*e*g^2*n*q*x*ln(c*x^n) 
/h^2-5/18*b*e*g*n*q*x^2*(a+b*ln(c*x^n))/h-2/9*b*d*n*x^3*(a+b*ln(c*x^n))+4/ 
27*b*e*n*q*x^3*(a+b*ln(c*x^n))-1/3*e*g^2*q*x*(a+b*ln(c*x^n))^2/h^2+1/6*e*g 
*q*x^2*(a+b*ln(c*x^n))^2/h+1/3*d*x^3*(a+b*ln(c*x^n))^2-1/9*e*q*x^3*(a+b*ln 
(c*x^n))^2+2/27*b^2*e*g^3*n^2*q*ln(h*x+g)/h^3+2/27*b^2*e*n^2*x^3*ln(f*(h*x 
+g)^q)-2/9*b*e*n*x^3*(a+b*ln(c*x^n))*ln(f*(h*x+g)^q)+1/3*e*x^3*(a+b*ln(c*x 
^n))^2*ln(f*(h*x+g)^q)-2/9*b*e*g^3*n*q*(a+b*ln(c*x^n))*ln(1+h*x/g)/h^3+1/3 
*e*g^3*q*(a+b*ln(c*x^n))^2*ln(1+h*x/g)/h^3-2/9*b^2*e*g^3*n^2*q*polylog(2,- 
h*x/g)/h^3+2/3*b*e*g^3*n*q*(a+b*ln(c*x^n))*polylog(2,-h*x/g)/h^3-2/3*b^2*e 
*g^3*n^2*q*polylog(3,-h*x/g)/h^3
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 923, normalized size of antiderivative = 1.76 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(-36*a^2*e*g^2*h*q*x + 96*a*b*e*g^2*h*n*q*x - 104*b^2*e*g^2*h*n^2*q*x + 18 
*a^2*e*g*h^2*q*x^2 - 30*a*b*e*g*h^2*n*q*x^2 + 19*b^2*e*g*h^2*n^2*q*x^2 + 3 
6*a^2*d*h^3*x^3 - 24*a*b*d*h^3*n*x^3 + 8*b^2*d*h^3*n^2*x^3 - 12*a^2*e*h^3* 
q*x^3 + 16*a*b*e*h^3*n*q*x^3 - 8*b^2*e*h^3*n^2*q*x^3 - 72*a*b*e*g^2*h*q*x* 
Log[c*x^n] + 96*b^2*e*g^2*h*n*q*x*Log[c*x^n] + 36*a*b*e*g*h^2*q*x^2*Log[c* 
x^n] - 30*b^2*e*g*h^2*n*q*x^2*Log[c*x^n] + 72*a*b*d*h^3*x^3*Log[c*x^n] - 2 
4*b^2*d*h^3*n*x^3*Log[c*x^n] - 24*a*b*e*h^3*q*x^3*Log[c*x^n] + 16*b^2*e*h^ 
3*n*q*x^3*Log[c*x^n] - 36*b^2*e*g^2*h*q*x*Log[c*x^n]^2 + 18*b^2*e*g*h^2*q* 
x^2*Log[c*x^n]^2 + 36*b^2*d*h^3*x^3*Log[c*x^n]^2 - 12*b^2*e*h^3*q*x^3*Log[ 
c*x^n]^2 + 36*a^2*e*g^3*q*Log[g + h*x] - 24*a*b*e*g^3*n*q*Log[g + h*x] + 8 
*b^2*e*g^3*n^2*q*Log[g + h*x] - 72*a*b*e*g^3*n*q*Log[x]*Log[g + h*x] + 24* 
b^2*e*g^3*n^2*q*Log[x]*Log[g + h*x] + 36*b^2*e*g^3*n^2*q*Log[x]^2*Log[g + 
h*x] + 72*a*b*e*g^3*q*Log[c*x^n]*Log[g + h*x] - 24*b^2*e*g^3*n*q*Log[c*x^n 
]*Log[g + h*x] - 72*b^2*e*g^3*n*q*Log[x]*Log[c*x^n]*Log[g + h*x] + 36*b^2* 
e*g^3*q*Log[c*x^n]^2*Log[g + h*x] + 36*a^2*e*h^3*x^3*Log[f*(g + h*x)^q] - 
24*a*b*e*h^3*n*x^3*Log[f*(g + h*x)^q] + 8*b^2*e*h^3*n^2*x^3*Log[f*(g + h*x 
)^q] + 72*a*b*e*h^3*x^3*Log[c*x^n]*Log[f*(g + h*x)^q] - 24*b^2*e*h^3*n*x^3 
*Log[c*x^n]*Log[f*(g + h*x)^q] + 36*b^2*e*h^3*x^3*Log[c*x^n]^2*Log[f*(g + 
h*x)^q] + 72*a*b*e*g^3*n*q*Log[x]*Log[1 + (h*x)/g] - 24*b^2*e*g^3*n^2*q*Lo 
g[x]*Log[1 + (h*x)/g] - 36*b^2*e*g^3*n^2*q*Log[x]^2*Log[1 + (h*x)/g] + ...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2891

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2891

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))^(q_.)*((k_.) + (l_.)*(x_))^(r_.), 
 x_Symbol] :> Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^p*(f + g* 
Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, 
n, p, q, r}, x]
 

Maple [C] (warning: unable to verify)

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

Time = 0.09 (sec) , antiderivative size = 6097, normalized size of antiderivative = 11.61

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/3*b^2*d*x^3*log(c*x^n)^2 - 2/9*a*b*d*n*x^3 + 1/3*a^2*e*x^3*log((h*x + g) 
^q*f) + 2/3*a*b*d*x^3*log(c*x^n) + 1/3*a^2*d*x^3 + 1/18*a^2*e*h*q*(6*g^3*l 
og(h*x + g)/h^4 - (2*h^2*x^3 - 3*g*h*x^2 + 6*g^2*x)/h^3) + 2/27*(n^2*x^3 - 
 3*n*x^3*log(c*x^n))*b^2*d + 1/54*(3*(3*b^2*e*g*h^2*q*x^2 - 6*b^2*e*g^2*h* 
q*x + 6*b^2*e*g^3*q*log(h*x + g) - 2*(h^3*q - 3*h^3*log(f))*b^2*e*x^3)*log 
(x^n)^2 + 2*(9*b^2*e*h^3*x^3*log(x^n)^2 + 6*(3*a*b*e*h^3 - (e*h^3*n - 3*e* 
h^3*log(c))*b^2)*x^3*log(x^n) - (6*(e*h^3*n - 3*e*h^3*log(c))*a*b - (2*e*h 
^3*n^2 - 6*e*h^3*n*log(c) + 9*e*h^3*log(c)^2)*b^2)*x^3)*log((h*x + g)^q))/ 
h^3 - integrate(-1/27*((6*(e*h^4*n*q - 3*(h^4*q - 3*h^4*log(f))*e*log(c))* 
a*b - (2*e*h^4*n^2*q - 6*e*h^4*n*q*log(c) + 9*(h^4*q - 3*h^4*log(f))*e*log 
(c)^2)*b^2)*x^4 + 27*(b^2*e*g*h^3*log(c)^2*log(f) + 2*a*b*e*g*h^3*log(c)*l 
og(f))*x^3 + 3*(3*b^2*e*g^2*h^2*n*q*x^2 + 6*b^2*e*g^3*h*n*q*x - 2*(3*(h^4* 
q - 3*h^4*log(f))*a*b*e + (3*(h^4*q - 3*h^4*log(f))*e*log(c) - (2*h^4*n*q 
- 3*h^4*n*log(f))*e)*b^2)*x^4 + (18*a*b*e*g*h^3*log(f) + (18*e*g*h^3*log(c 
)*log(f) - (g*h^3*n*q + 6*g*h^3*n*log(f))*e)*b^2)*x^3 - 6*(b^2*e*g^3*h*n*q 
*x + b^2*e*g^4*n*q)*log(h*x + g))*log(x^n))/(h^4*x^2 + g*h^3*x), x)
 

Giac [F]

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

integrate(x^2*(a+b*log(c*x^n))^2*(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)^2*x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 36*int(log(x**n*c)**2/(g*x + h*x**2),x)*b**2*e*g**4*n*q - 72*int(log(x 
**n*c)/(g*x + h*x**2),x)*a*b*e*g**4*n*q + 24*int(log(x**n*c)/(g*x + h*x**2 
),x)*b**2*e*g**4*n**2*q + 36*log((g + h*x)**q*f)*log(x**n*c)**2*b**2*e*h** 
3*n*x**3 + 72*log((g + h*x)**q*f)*log(x**n*c)*a*b*e*h**3*n*x**3 - 24*log(( 
g + h*x)**q*f)*log(x**n*c)*b**2*e*h**3*n**2*x**3 + 36*log((g + h*x)**q*f)* 
a**2*e*g**3*n + 36*log((g + h*x)**q*f)*a**2*e*h**3*n*x**3 - 24*log((g + h* 
x)**q*f)*a*b*e*g**3*n**2 - 24*log((g + h*x)**q*f)*a*b*e*h**3*n**2*x**3 + 8 
*log((g + h*x)**q*f)*b**2*e*g**3*n**3 + 8*log((g + h*x)**q*f)*b**2*e*h**3* 
n**3*x**3 + 12*log(x**n*c)**3*b**2*e*g**3*q + 36*log(x**n*c)**2*a*b*e*g**3 
*q + 36*log(x**n*c)**2*b**2*d*h**3*n*x**3 - 12*log(x**n*c)**2*b**2*e*g**3* 
n*q - 36*log(x**n*c)**2*b**2*e*g**2*h*n*q*x + 18*log(x**n*c)**2*b**2*e*g*h 
**2*n*q*x**2 - 12*log(x**n*c)**2*b**2*e*h**3*n*q*x**3 + 72*log(x**n*c)*a*b 
*d*h**3*n*x**3 - 72*log(x**n*c)*a*b*e*g**2*h*n*q*x + 36*log(x**n*c)*a*b*e* 
g*h**2*n*q*x**2 - 24*log(x**n*c)*a*b*e*h**3*n*q*x**3 - 24*log(x**n*c)*b**2 
*d*h**3*n**2*x**3 + 96*log(x**n*c)*b**2*e*g**2*h*n**2*q*x - 30*log(x**n*c) 
*b**2*e*g*h**2*n**2*q*x**2 + 16*log(x**n*c)*b**2*e*h**3*n**2*q*x**3 + 36*a 
**2*d*h**3*n*x**3 - 36*a**2*e*g**2*h*n*q*x + 18*a**2*e*g*h**2*n*q*x**2 - 1 
2*a**2*e*h**3*n*q*x**3 - 24*a*b*d*h**3*n**2*x**3 + 96*a*b*e*g**2*h*n**2*q* 
x - 30*a*b*e*g*h**2*n**2*q*x**2 + 16*a*b*e*h**3*n**2*q*x**3 + 8*b**2*d*h** 
3*n**3*x**3 - 104*b**2*e*g**2*h*n**3*q*x + 19*b**2*e*g*h**2*n**3*q*x**2...