\(\int (s+t x)^2 (a+b \log (c x^n))^2 (d+e \log (f (g+h x)^q)) \, dx\) [418]

Optimal result

Integrand size = 34, antiderivative size = 1454 \[ \int (s+t x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx =\text {Too large to display} \] Output:

b*e*g^2*n*q*s*t*(a+b*ln(c*x^n))*ln(1+h*x/g)/h^2+e*s*t*x^2*(a+b*ln(c*x^n))^ 
2*ln(f*(h*x+g)^q)+e*s^2*x*(a+b*ln(c*x^n))^2*ln(f*(h*x+g)^q)-e*q*s^2*x*(a+b 
*ln(c*x^n))^2-2/3*b*d*n*s^3*ln(x)*(a+b*ln(c*x^n))/t-1/3*e*g^2*q*t^2*x*(a+b 
*ln(c*x^n))^2/h^2+1/6*e*g*q*t^2*x^2*(a+b*ln(c*x^n))^2/h-2*b^2*e*n*s^2*x*ln 
(c*x^n)*ln(f*(h*x+g)^q)-2/9*b*e*n*t^2*x^3*(a+b*ln(c*x^n))*ln(f*(h*x+g)^q)+ 
1/3*e*g^3*q*t^2*(a+b*ln(c*x^n))^2*ln(1+h*x/g)/h^3+4*b^2*e*n*q*s^2*x*ln(c*x 
^n)+4/27*b*e*n*q*t^2*x^3*(a+b*ln(c*x^n))-2*a*b*e*n*s^2*x*ln(f*(h*x+g)^q)+1 
/2*b^2*e*n^2*s*t*x^2*ln(f*(h*x+g)^q)+2*a*b*e*n*q*s^2*x+2*b*e*n*(-b*n+a)*q* 
s^2*x-3/4*b^2*e*n^2*q*s*t*x^2+2*b^2*d*n^2*s^2*x+2/27*b^2*d*n^2*t^2*x^3-1/9 
*e*q*t^2*x^3*(a+b*ln(c*x^n))^2-5/18*b*e*g*n*q*t^2*x^2*(a+b*ln(c*x^n))/h+7/ 
2*b^2*e*g*n^2*q*s*t*x/h+8/9*a*b*e*g^2*n*q*t^2*x/h^2+1/3*e*t^2*x^3*(a+b*ln( 
c*x^n))^2*ln(f*(h*x+g)^q)-26/27*b^2*e*g^2*n^2*q*t^2*x/h^2+19/108*b^2*e*g*n 
^2*q*t^2*x^2/h-2*b^2*e*g*n^2*q*s^2*polylog(3,-h*x/g)/h-2/3*b^2*e*g^3*n^2*q 
*t^2*polylog(3,-h*x/g)/h^3-2*b^2*e*g*n^2*q*s^2*polylog(2,-h*x/g)/h-2/9*b^2 
*e*g^3*n^2*q*t^2*polylog(2,-h*x/g)/h^3+2/27*b^2*e*g^3*n^2*q*t^2*ln(h*x+g)/ 
h^3+2*b^2*e*g^2*n^2*q*s*t*polylog(3,-h*x/g)/h^2-1/2*b^2*e*g^2*n^2*q*s*t*ln 
(h*x+g)/h^2-2*b*e*g*n*(-b*n+a)*q*s^2*ln(h*x+g)/h+2*b*e*g*n*q*s^2*(a+b*ln(c 
*x^n))*polylog(2,-h*x/g)/h+2/3*b*e*g^3*n*q*t^2*(a+b*ln(c*x^n))*polylog(2,- 
h*x/g)/h^3-2*b^2*e*g*n*q*s^2*ln(c*x^n)*ln(1+h*x/g)/h-2/9*b*e*g^3*n*q*t^2*( 
a+b*ln(c*x^n))*ln(1+h*x/g)/h^3+8/9*b^2*e*g^2*n*q*t^2*x*ln(c*x^n)/h^2-3*...
 

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 2716, normalized size of antiderivative = 1.87 \[ \int (s+t x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\text {Result too large to show} \] Input:

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

Output:

a^2*d*s^2*x - 2*a*b*d*n*s^2*x + 2*b^2*d*n^2*s^2*x - a^2*e*q*s^2*x + 4*a*b* 
e*n*q*s^2*x - 6*b^2*e*n^2*q*s^2*x + (a^2*e*g*q*s*t*x)/h - (3*a*b*e*g*n*q*s 
*t*x)/h + (7*b^2*e*g*n^2*q*s*t*x)/(2*h) - (a^2*e*g^2*q*t^2*x)/(3*h^2) + (8 
*a*b*e*g^2*n*q*t^2*x)/(9*h^2) - (26*b^2*e*g^2*n^2*q*t^2*x)/(27*h^2) + a^2* 
d*s*t*x^2 - a*b*d*n*s*t*x^2 + (b^2*d*n^2*s*t*x^2)/2 - (a^2*e*q*s*t*x^2)/2 
+ a*b*e*n*q*s*t*x^2 - (3*b^2*e*n^2*q*s*t*x^2)/4 + (a^2*e*g*q*t^2*x^2)/(6*h 
) - (5*a*b*e*g*n*q*t^2*x^2)/(18*h) + (19*b^2*e*g*n^2*q*t^2*x^2)/(108*h) + 
(a^2*d*t^2*x^3)/3 - (2*a*b*d*n*t^2*x^3)/9 + (2*b^2*d*n^2*t^2*x^3)/27 - (a^ 
2*e*q*t^2*x^3)/9 + (4*a*b*e*n*q*t^2*x^3)/27 - (2*b^2*e*n^2*q*t^2*x^3)/27 + 
 2*a*b*d*s^2*x*Log[c*x^n] - 2*b^2*d*n*s^2*x*Log[c*x^n] - 2*a*b*e*q*s^2*x*L 
og[c*x^n] + 4*b^2*e*n*q*s^2*x*Log[c*x^n] + (2*a*b*e*g*q*s*t*x*Log[c*x^n])/ 
h - (3*b^2*e*g*n*q*s*t*x*Log[c*x^n])/h - (2*a*b*e*g^2*q*t^2*x*Log[c*x^n])/ 
(3*h^2) + (8*b^2*e*g^2*n*q*t^2*x*Log[c*x^n])/(9*h^2) + 2*a*b*d*s*t*x^2*Log 
[c*x^n] - b^2*d*n*s*t*x^2*Log[c*x^n] - a*b*e*q*s*t*x^2*Log[c*x^n] + b^2*e* 
n*q*s*t*x^2*Log[c*x^n] + (a*b*e*g*q*t^2*x^2*Log[c*x^n])/(3*h) - (5*b^2*e*g 
*n*q*t^2*x^2*Log[c*x^n])/(18*h) + (2*a*b*d*t^2*x^3*Log[c*x^n])/3 - (2*b^2* 
d*n*t^2*x^3*Log[c*x^n])/9 - (2*a*b*e*q*t^2*x^3*Log[c*x^n])/9 + (4*b^2*e*n* 
q*t^2*x^3*Log[c*x^n])/27 + b^2*d*s^2*x*Log[c*x^n]^2 - b^2*e*q*s^2*x*Log[c* 
x^n]^2 + (b^2*e*g*q*s*t*x*Log[c*x^n]^2)/h - (b^2*e*g^2*q*t^2*x*Log[c*x^n]^ 
2)/(3*h^2) + b^2*d*s*t*x^2*Log[c*x^n]^2 - (b^2*e*q*s*t*x^2*Log[c*x^n]^2...
 

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.

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

\(\Big \downarrow \) 2891

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

Input:

Int[(s + t*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 = 0.58 (sec) , antiderivative size = 16290, normalized size of antiderivative = 11.20

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

Input:

int((t*x+s)^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]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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