Integrand size = 28, antiderivative size = 443 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=-\frac {3 a b e g n q x}{2 h}+\frac {7 b^2 e g n^2 q x}{4 h}+\frac {1}{4} b^2 d n^2 x^2-\frac {3}{8} b^2 e n^2 q x^2-\frac {3 b^2 e g n q x \log \left (c x^n\right )}{2 h}-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} b e n q x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {e g q x \left (a+b \log \left (c x^n\right )\right )^2}{2 h}+\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} e q x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 e g^2 n^2 q \log (g+h x)}{4 h^2}+\frac {1}{4} b^2 e n^2 x^2 \log \left (f (g+h x)^q\right )-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (f (g+h x)^q\right )+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (f (g+h x)^q\right )+\frac {b e g^2 n q \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {h x}{g}\right )}{2 h^2}-\frac {e g^2 q \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {h x}{g}\right )}{2 h^2}+\frac {b^2 e g^2 n^2 q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{2 h^2}-\frac {b e g^2 n q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^2}+\frac {b^2 e g^2 n^2 q \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right )}{h^2} \] Output:
-3/2*a*b*e*g*n*q*x/h+7/4*b^2*e*g*n^2*q*x/h+1/4*b^2*d*n^2*x^2-3/8*b^2*e*n^2 *q*x^2-3/2*b^2*e*g*n*q*x*ln(c*x^n)/h-1/2*b*d*n*x^2*(a+b*ln(c*x^n))+1/2*b*e *n*q*x^2*(a+b*ln(c*x^n))+1/2*e*g*q*x*(a+b*ln(c*x^n))^2/h+1/2*d*x^2*(a+b*ln (c*x^n))^2-1/4*e*q*x^2*(a+b*ln(c*x^n))^2-1/4*b^2*e*g^2*n^2*q*ln(h*x+g)/h^2 +1/4*b^2*e*n^2*x^2*ln(f*(h*x+g)^q)-1/2*b*e*n*x^2*(a+b*ln(c*x^n))*ln(f*(h*x +g)^q)+1/2*e*x^2*(a+b*ln(c*x^n))^2*ln(f*(h*x+g)^q)+1/2*b*e*g^2*n*q*(a+b*ln (c*x^n))*ln(1+h*x/g)/h^2-1/2*e*g^2*q*(a+b*ln(c*x^n))^2*ln(1+h*x/g)/h^2+1/2 *b^2*e*g^2*n^2*q*polylog(2,-h*x/g)/h^2-b*e*g^2*n*q*(a+b*ln(c*x^n))*polylog (2,-h*x/g)/h^2+b^2*e*g^2*n^2*q*polylog(3,-h*x/g)/h^2
Time = 0.37 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.81 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {4 a^2 e g h q x-12 a b e g h n q x+14 b^2 e g h n^2 q x+4 a^2 d h^2 x^2-4 a b d h^2 n x^2+2 b^2 d h^2 n^2 x^2-2 a^2 e h^2 q x^2+4 a b e h^2 n q x^2-3 b^2 e h^2 n^2 q x^2+8 a b e g h q x \log \left (c x^n\right )-12 b^2 e g h n q x \log \left (c x^n\right )+8 a b d h^2 x^2 \log \left (c x^n\right )-4 b^2 d h^2 n x^2 \log \left (c x^n\right )-4 a b e h^2 q x^2 \log \left (c x^n\right )+4 b^2 e h^2 n q x^2 \log \left (c x^n\right )+4 b^2 e g h q x \log ^2\left (c x^n\right )+4 b^2 d h^2 x^2 \log ^2\left (c x^n\right )-2 b^2 e h^2 q x^2 \log ^2\left (c x^n\right )-4 a^2 e g^2 q \log (g+h x)+4 a b e g^2 n q \log (g+h x)-2 b^2 e g^2 n^2 q \log (g+h x)+8 a b e g^2 n q \log (x) \log (g+h x)-4 b^2 e g^2 n^2 q \log (x) \log (g+h x)-4 b^2 e g^2 n^2 q \log ^2(x) \log (g+h x)-8 a b e g^2 q \log \left (c x^n\right ) \log (g+h x)+4 b^2 e g^2 n q \log \left (c x^n\right ) \log (g+h x)+8 b^2 e g^2 n q \log (x) \log \left (c x^n\right ) \log (g+h x)-4 b^2 e g^2 q \log ^2\left (c x^n\right ) \log (g+h x)+4 a^2 e h^2 x^2 \log \left (f (g+h x)^q\right )-4 a b e h^2 n x^2 \log \left (f (g+h x)^q\right )+2 b^2 e h^2 n^2 x^2 \log \left (f (g+h x)^q\right )+8 a b e h^2 x^2 \log \left (c x^n\right ) \log \left (f (g+h x)^q\right )-4 b^2 e h^2 n x^2 \log \left (c x^n\right ) \log \left (f (g+h x)^q\right )+4 b^2 e h^2 x^2 \log ^2\left (c x^n\right ) \log \left (f (g+h x)^q\right )-8 a b e g^2 n q \log (x) \log \left (1+\frac {h x}{g}\right )+4 b^2 e g^2 n^2 q \log (x) \log \left (1+\frac {h x}{g}\right )+4 b^2 e g^2 n^2 q \log ^2(x) \log \left (1+\frac {h x}{g}\right )-8 b^2 e g^2 n q \log (x) \log \left (c x^n\right ) \log \left (1+\frac {h x}{g}\right )+4 b e g^2 n q \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )+8 b^2 e g^2 n^2 q \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right )}{8 h^2} \] Input:
Integrate[x*(a + b*Log[c*x^n])^2*(d + e*Log[f*(g + h*x)^q]),x]
Output:
(4*a^2*e*g*h*q*x - 12*a*b*e*g*h*n*q*x + 14*b^2*e*g*h*n^2*q*x + 4*a^2*d*h^2 *x^2 - 4*a*b*d*h^2*n*x^2 + 2*b^2*d*h^2*n^2*x^2 - 2*a^2*e*h^2*q*x^2 + 4*a*b *e*h^2*n*q*x^2 - 3*b^2*e*h^2*n^2*q*x^2 + 8*a*b*e*g*h*q*x*Log[c*x^n] - 12*b ^2*e*g*h*n*q*x*Log[c*x^n] + 8*a*b*d*h^2*x^2*Log[c*x^n] - 4*b^2*d*h^2*n*x^2 *Log[c*x^n] - 4*a*b*e*h^2*q*x^2*Log[c*x^n] + 4*b^2*e*h^2*n*q*x^2*Log[c*x^n ] + 4*b^2*e*g*h*q*x*Log[c*x^n]^2 + 4*b^2*d*h^2*x^2*Log[c*x^n]^2 - 2*b^2*e* h^2*q*x^2*Log[c*x^n]^2 - 4*a^2*e*g^2*q*Log[g + h*x] + 4*a*b*e*g^2*n*q*Log[ g + h*x] - 2*b^2*e*g^2*n^2*q*Log[g + h*x] + 8*a*b*e*g^2*n*q*Log[x]*Log[g + h*x] - 4*b^2*e*g^2*n^2*q*Log[x]*Log[g + h*x] - 4*b^2*e*g^2*n^2*q*Log[x]^2 *Log[g + h*x] - 8*a*b*e*g^2*q*Log[c*x^n]*Log[g + h*x] + 4*b^2*e*g^2*n*q*Lo g[c*x^n]*Log[g + h*x] + 8*b^2*e*g^2*n*q*Log[x]*Log[c*x^n]*Log[g + h*x] - 4 *b^2*e*g^2*q*Log[c*x^n]^2*Log[g + h*x] + 4*a^2*e*h^2*x^2*Log[f*(g + h*x)^q ] - 4*a*b*e*h^2*n*x^2*Log[f*(g + h*x)^q] + 2*b^2*e*h^2*n^2*x^2*Log[f*(g + h*x)^q] + 8*a*b*e*h^2*x^2*Log[c*x^n]*Log[f*(g + h*x)^q] - 4*b^2*e*h^2*n*x^ 2*Log[c*x^n]*Log[f*(g + h*x)^q] + 4*b^2*e*h^2*x^2*Log[c*x^n]^2*Log[f*(g + h*x)^q] - 8*a*b*e*g^2*n*q*Log[x]*Log[1 + (h*x)/g] + 4*b^2*e*g^2*n^2*q*Log[ x]*Log[1 + (h*x)/g] + 4*b^2*e*g^2*n^2*q*Log[x]^2*Log[1 + (h*x)/g] - 8*b^2* e*g^2*n*q*Log[x]*Log[c*x^n]*Log[1 + (h*x)/g] + 4*b*e*g^2*n*q*(-2*a + b*n - 2*b*Log[c*x^n])*PolyLog[2, -((h*x)/g)] + 8*b^2*e*g^2*n^2*q*PolyLog[3, -(( h*x)/g)])/(8*h^2)
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.
| \(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2891 |
| \(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )dx\) |
Input:
Int[x*(a + b*Log[c*x^n])^2*(d + e*Log[f*(g + h*x)^q]),x]
Output:
$Aborted
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 226.20 (sec) , antiderivative size = 4990, normalized size of antiderivative = 11.26
Input:
int(x*(a+b*ln(c*x^n))^2*(d+e*ln(f*(h*x+g)^q)),x,method=_RETURNVERBOSE)
Output:
-1/2*q*b*e*ln(x^n)*x^2*a+1/2*q*b*e*n*x^2*a-1/2*q*b^2*e*ln(x^n)*x^2*ln(c)+1 /2*q*b^2*e*n*x^2*ln(c)+1/2*q*e*b^2*n*ln(x^n)*x^2+1/16*q*e*x^2*Pi^2*b^2*csg n(I*c*x^n)^6-1/2*q*g^2*a^2*e/h^2*ln(h*x+g)-1/4*q*e*b^2*ln(x^n)^2*x^2-1/2*q *e*x^2*ln(c)*a*b+1/2/h*q*e*g*x*a^2-1/4*q*e*x^2*a^2-1/8/h*q*e*g*x*Pi^2*b^2* csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+1/8*q/h^2*e*g^2*ln(h*x+g)*Pi^2*b ^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+1/2*q/h^2*e*g^2*ln(h*x+g)*Pi^ 2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-1/4*q/h^2*e*g^2*ln(h*x+g)*Pi^2 *b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-1/2/h*q*e*g*x*Pi^2*b^2*csgn(I *x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/4/h*q*e*g*x*Pi^2*b^2*csgn(I*x^n)*csgn(I* c*x^n)^3*csgn(I*c)^2-3/2*a*b*e*g*n*q*x/h+(1/2*x^2*b^2*e*ln(x^n)^2+1/2*b*e* x^2*(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)*c sgn(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) -n*b+2*a)*ln(x^n)+1/8*e*x^2*(2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn (I*c)-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^2*csgn(I *x^n)*csgn(I*c*x^n)^4*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I *Pi*ln(c)*b^2*csgn(I*c*x^n)^2*csgn(I*c)-4*b^2*ln(c)*n+8*a*b*ln(c)+4*a^2+4* I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)+4*b^2*ln(c)^2+2*b^2*n^2-4*I*Pi*ln(c)*b^ 2*csgn(I*c*x^n)^3+2*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)-Pi^2*b^2*csgn(I*c*x ^n)^4*csgn(I*c)^2-Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*Pi^2*b^2*csgn(I *x^n)*csgn(I*c*x^n)^5-4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2*...
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int { {\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q)),x, algorithm="fricas ")
Output:
integral(b^2*d*x*log(c*x^n)^2 + 2*a*b*d*x*log(c*x^n) + a^2*d*x + (b^2*e*x* log(c*x^n)^2 + 2*a*b*e*x*log(c*x^n) + a^2*e*x)*log((h*x + g)^q*f), x)
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*ln(c*x**n))**2*(d+e*ln(f*(h*x+g)**q)),x)
Output:
Timed out
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int { {\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q)),x, algorithm="maxima ")
Output:
1/2*b^2*d*x^2*log(c*x^n)^2 - 1/2*a*b*d*n*x^2 - 1/4*a^2*e*h*q*(2*g^2*log(h* x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + 1/2*a^2*e*x^2*log((h*x + g)^q*f) + a*b *d*x^2*log(c*x^n) + 1/2*a^2*d*x^2 + 1/4*(n^2*x^2 - 2*n*x^2*log(c*x^n))*b^2 *d + 1/4*((2*b^2*e*g*h*q*x - 2*b^2*e*g^2*q*log(h*x + g) - (h^2*q - 2*h^2*l og(f))*b^2*e*x^2)*log(x^n)^2 + (2*b^2*e*h^2*x^2*log(x^n)^2 + 2*(2*a*b*e*h^ 2 - (e*h^2*n - 2*e*h^2*log(c))*b^2)*x^2*log(x^n) - (2*(e*h^2*n - 2*e*h^2*l og(c))*a*b - (e*h^2*n^2 - 2*e*h^2*n*log(c) + 2*e*h^2*log(c)^2)*b^2)*x^2)*l og((h*x + g)^q))/h^2 + integrate(1/4*((2*(e*h^3*n*q - 2*(h^3*q - 2*h^3*log (f))*e*log(c))*a*b - (e*h^3*n^2*q - 2*e*h^3*n*q*log(c) + 2*(h^3*q - 2*h^3* log(f))*e*log(c)^2)*b^2)*x^3 + 4*(b^2*e*g*h^2*log(c)^2*log(f) + 2*a*b*e*g* h^2*log(c)*log(f))*x^2 - 2*(2*b^2*e*g^2*h*n*q*x + 2*((h^3*q - 2*h^3*log(f) )*a*b*e + ((h^3*q - 2*h^3*log(f))*e*log(c) - (h^3*n*q - h^3*n*log(f))*e)*b ^2)*x^3 - (4*a*b*e*g*h^2*log(f) + (4*e*g*h^2*log(c)*log(f) - (g*h^2*n*q + 2*g*h^2*n*log(f))*e)*b^2)*x^2 - 2*(b^2*e*g^2*h*n*q*x + b^2*e*g^3*n*q)*log( h*x + g))*log(x^n))/(h^3*x^2 + g*h^2*x), x)
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int { {\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(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, x)
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int x\,\left (d+e\,\ln \left (f\,{\left (g+h\,x\right )}^q\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:
int(x*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n))^2,x)
Output:
int(x*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n))^2, x)
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx =\text {Too large to display} \] Input:
int(x*(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 + 24*int(log(x**n *c)/(g*x + h*x**2),x)*a*b*e*g**3*n*q - 12*int(log(x**n*c)/(g*x + h*x**2),x )*b**2*e*g**3*n**2*q + 12*log((g + h*x)**q*f)*log(x**n*c)**2*b**2*e*h**2*n *x**2 + 24*log((g + h*x)**q*f)*log(x**n*c)*a*b*e*h**2*n*x**2 - 12*log((g + h*x)**q*f)*log(x**n*c)*b**2*e*h**2*n**2*x**2 - 12*log((g + h*x)**q*f)*a** 2*e*g**2*n + 12*log((g + h*x)**q*f)*a**2*e*h**2*n*x**2 + 12*log((g + h*x)* *q*f)*a*b*e*g**2*n**2 - 12*log((g + h*x)**q*f)*a*b*e*h**2*n**2*x**2 - 6*lo g((g + h*x)**q*f)*b**2*e*g**2*n**3 + 6*log((g + h*x)**q*f)*b**2*e*h**2*n** 3*x**2 - 4*log(x**n*c)**3*b**2*e*g**2*q - 12*log(x**n*c)**2*a*b*e*g**2*q + 12*log(x**n*c)**2*b**2*d*h**2*n*x**2 + 6*log(x**n*c)**2*b**2*e*g**2*n*q + 12*log(x**n*c)**2*b**2*e*g*h*n*q*x - 6*log(x**n*c)**2*b**2*e*h**2*n*q*x** 2 + 24*log(x**n*c)*a*b*d*h**2*n*x**2 + 24*log(x**n*c)*a*b*e*g*h*n*q*x - 12 *log(x**n*c)*a*b*e*h**2*n*q*x**2 - 12*log(x**n*c)*b**2*d*h**2*n**2*x**2 - 36*log(x**n*c)*b**2*e*g*h*n**2*q*x + 12*log(x**n*c)*b**2*e*h**2*n**2*q*x** 2 + 12*a**2*d*h**2*n*x**2 + 12*a**2*e*g*h*n*q*x - 6*a**2*e*h**2*n*q*x**2 - 12*a*b*d*h**2*n**2*x**2 - 36*a*b*e*g*h*n**2*q*x + 12*a*b*e*h**2*n**2*q*x* *2 + 6*b**2*d*h**2*n**3*x**2 + 42*b**2*e*g*h*n**3*q*x - 9*b**2*e*h**2*n**3 *q*x**2)/(24*h**2*n)