\(\int (s+t x) (a+b \log (c x^n))^2 (d+e \log (f (g+h x)^q)) \, dx\) [417]

Optimal result

Integrand size = 32, antiderivative size = 829 \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx =\text {Too large to display} \] Output:

2*a*b*e*n*q*s*x+2*b*e*n*(-b*n+a)*q*s*x-2*a*b*e*n*s*x*ln(f*(h*x+g)^q)+4*b^2 
*e*n*q*s*x*ln(c*x^n)+1/2*b*e*n*q*t*x^2*(a+b*ln(c*x^n))+1/2*e*g*q*t*x*(a+b* 
ln(c*x^n))^2/h-2*b^2*e*n*s*x*ln(c*x^n)*ln(f*(h*x+g)^q)-1/2*b*e*n*t*x^2*(a+ 
b*ln(c*x^n))*ln(f*(h*x+g)^q)-1/2*e*g^2*q*t*(a+b*ln(c*x^n))^2*ln(1+h*x/g)/h 
^2+2*b^2*d*n^2*s*x+1/4*b^2*d*n^2*t*x^2-1/4*e*q*t*x^2*(a+b*ln(c*x^n))^2+1/2 
*e*t*x^2*(a+b*ln(c*x^n))^2*ln(f*(h*x+g)^q)+e*s*x*(a+b*ln(c*x^n))^2*ln(f*(h 
*x+g)^q)-e*q*s*x*(a+b*ln(c*x^n))^2+d*s*x*(a+b*ln(c*x^n))^2-2*b^2*e*g*n^2*q 
*s*polylog(3,-h*x/g)/h-2*b^2*e*g*n^2*q*s*polylog(2,-h*x/g)/h+1/2*b^2*e*g^2 
*n^2*q*t*polylog(2,-h*x/g)/h^2-1/4*b^2*e*g^2*n^2*q*t*ln(h*x+g)/h^2+7/4*b^2 
*e*g*n^2*q*t*x/h-2*b^2*e*g*n*q*s*ln(c*x^n)*ln(1+h*x/g)/h+1/2*b*e*g^2*n*q*t 
*(a+b*ln(c*x^n))*ln(1+h*x/g)/h^2-3/2*a*b*e*g*n*q*t*x/h-2*b*e*g*n*(-b*n+a)* 
q*s*ln(h*x+g)/h-3/2*b^2*e*g*n*q*t*x*ln(c*x^n)/h+2*b*e*g*n*q*s*(a+b*ln(c*x^ 
n))*polylog(2,-h*x/g)/h+1/2*d*t*x^2*(a+b*ln(c*x^n))^2+b^2*e*g^2*n^2*q*t*po 
lylog(3,-h*x/g)/h^2+e*g*q*s*(a+b*ln(c*x^n))^2*ln(1+h*x/g)/h-b*e*g^2*n*q*t* 
(a+b*ln(c*x^n))*polylog(2,-h*x/g)/h^2-2*a*b*d*n*s*x-4*b^2*e*n^2*q*s*x-3/8* 
b^2*e*n^2*q*t*x^2+2*b^2*e*n^2*s*x*ln(f*(h*x+g)^q)+1/4*b^2*e*n^2*t*x^2*ln(f 
*(h*x+g)^q)-2*b^2*d*n*s*x*ln(c*x^n)-1/2*b*d*n*t*x^2*(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 1507, normalized size of antiderivative = 1.82 \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(8*a^2*d*h^2*s*x - 16*a*b*d*h^2*n*s*x + 16*b^2*d*h^2*n^2*s*x - 8*a^2*e*h^2 
*q*s*x + 32*a*b*e*h^2*n*q*s*x - 48*b^2*e*h^2*n^2*q*s*x + 4*a^2*e*g*h*q*t*x 
 - 12*a*b*e*g*h*n*q*t*x + 14*b^2*e*g*h*n^2*q*t*x + 4*a^2*d*h^2*t*x^2 - 4*a 
*b*d*h^2*n*t*x^2 + 2*b^2*d*h^2*n^2*t*x^2 - 2*a^2*e*h^2*q*t*x^2 + 4*a*b*e*h 
^2*n*q*t*x^2 - 3*b^2*e*h^2*n^2*q*t*x^2 + 16*a*b*d*h^2*s*x*Log[c*x^n] - 16* 
b^2*d*h^2*n*s*x*Log[c*x^n] - 16*a*b*e*h^2*q*s*x*Log[c*x^n] + 32*b^2*e*h^2* 
n*q*s*x*Log[c*x^n] + 8*a*b*e*g*h*q*t*x*Log[c*x^n] - 12*b^2*e*g*h*n*q*t*x*L 
og[c*x^n] + 8*a*b*d*h^2*t*x^2*Log[c*x^n] - 4*b^2*d*h^2*n*t*x^2*Log[c*x^n] 
- 4*a*b*e*h^2*q*t*x^2*Log[c*x^n] + 4*b^2*e*h^2*n*q*t*x^2*Log[c*x^n] + 8*b^ 
2*d*h^2*s*x*Log[c*x^n]^2 - 8*b^2*e*h^2*q*s*x*Log[c*x^n]^2 + 4*b^2*e*g*h*q* 
t*x*Log[c*x^n]^2 + 4*b^2*d*h^2*t*x^2*Log[c*x^n]^2 - 2*b^2*e*h^2*q*t*x^2*Lo 
g[c*x^n]^2 + 8*a^2*e*g*h*q*s*Log[g + h*x] - 16*a*b*e*g*h*n*q*s*Log[g + h*x 
] + 16*b^2*e*g*h*n^2*q*s*Log[g + h*x] - 4*a^2*e*g^2*q*t*Log[g + h*x] + 4*a 
*b*e*g^2*n*q*t*Log[g + h*x] - 2*b^2*e*g^2*n^2*q*t*Log[g + h*x] - 16*a*b*e* 
g*h*n*q*s*Log[x]*Log[g + h*x] + 16*b^2*e*g*h*n^2*q*s*Log[x]*Log[g + h*x] + 
 8*a*b*e*g^2*n*q*t*Log[x]*Log[g + h*x] - 4*b^2*e*g^2*n^2*q*t*Log[x]*Log[g 
+ h*x] + 8*b^2*e*g*h*n^2*q*s*Log[x]^2*Log[g + h*x] - 4*b^2*e*g^2*n^2*q*t*L 
og[x]^2*Log[g + h*x] + 16*a*b*e*g*h*q*s*Log[c*x^n]*Log[g + h*x] - 16*b^2*e 
*g*h*n*q*s*Log[c*x^n]*Log[g + h*x] - 8*a*b*e*g^2*q*t*Log[c*x^n]*Log[g + h* 
x] + 4*b^2*e*g^2*n*q*t*Log[c*x^n]*Log[g + h*x] - 16*b^2*e*g*h*n*q*s*Log...
 

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.

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

\(\Big \downarrow \) 2891

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

Input:

Int[(s + t*x)*(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 = 0.41 (sec) , antiderivative size = 8919, normalized size of antiderivative = 10.76

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

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(12*int(log(x**n*c)**2/(g*x + h*x**2),x)*b**2*e*g**3*n*q*t - 24*int(log(x* 
*n*c)**2/(g*x + h*x**2),x)*b**2*e*g**2*h*n*q*s + 24*int(log(x**n*c)/(g*x + 
 h*x**2),x)*a*b*e*g**3*n*q*t - 48*int(log(x**n*c)/(g*x + h*x**2),x)*a*b*e* 
g**2*h*n*q*s - 12*int(log(x**n*c)/(g*x + h*x**2),x)*b**2*e*g**3*n**2*q*t + 
 48*int(log(x**n*c)/(g*x + h*x**2),x)*b**2*e*g**2*h*n**2*q*s + 24*log((g + 
 h*x)**q*f)*log(x**n*c)**2*b**2*e*h**2*n*s*x + 12*log((g + h*x)**q*f)*log( 
x**n*c)**2*b**2*e*h**2*n*t*x**2 + 48*log((g + h*x)**q*f)*log(x**n*c)*a*b*e 
*h**2*n*s*x + 24*log((g + h*x)**q*f)*log(x**n*c)*a*b*e*h**2*n*t*x**2 - 48* 
log((g + h*x)**q*f)*log(x**n*c)*b**2*e*h**2*n**2*s*x - 12*log((g + h*x)**q 
*f)*log(x**n*c)*b**2*e*h**2*n**2*t*x**2 - 12*log((g + h*x)**q*f)*a**2*e*g* 
*2*n*t + 24*log((g + h*x)**q*f)*a**2*e*g*h*n*s + 24*log((g + h*x)**q*f)*a* 
*2*e*h**2*n*s*x + 12*log((g + h*x)**q*f)*a**2*e*h**2*n*t*x**2 + 12*log((g 
+ h*x)**q*f)*a*b*e*g**2*n**2*t - 48*log((g + h*x)**q*f)*a*b*e*g*h*n**2*s - 
 48*log((g + h*x)**q*f)*a*b*e*h**2*n**2*s*x - 12*log((g + h*x)**q*f)*a*b*e 
*h**2*n**2*t*x**2 - 6*log((g + h*x)**q*f)*b**2*e*g**2*n**3*t + 48*log((g + 
 h*x)**q*f)*b**2*e*g*h*n**3*s + 48*log((g + h*x)**q*f)*b**2*e*h**2*n**3*s* 
x + 6*log((g + h*x)**q*f)*b**2*e*h**2*n**3*t*x**2 - 4*log(x**n*c)**3*b**2* 
e*g**2*q*t + 8*log(x**n*c)**3*b**2*e*g*h*q*s - 12*log(x**n*c)**2*a*b*e*g** 
2*q*t + 24*log(x**n*c)**2*a*b*e*g*h*q*s + 24*log(x**n*c)**2*b**2*d*h**2*n* 
s*x + 12*log(x**n*c)**2*b**2*d*h**2*n*t*x**2 + 6*log(x**n*c)**2*b**2*e*...