Integrand size = 30, antiderivative size = 400 \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=-\frac {1}{2} b d n s x+\frac {3}{4} b e n q s x-\frac {a e q (h s-g t) x}{2 h}+\frac {3 b e n q (h s-g t) x}{4 h}+\frac {b e n q (s+t x)^2}{4 t}+\frac {b e n q s^2 \log (x)}{4 t}-\frac {b e q (h s-g t) x \log \left (c x^n\right )}{2 h}-\frac {e q (s+t x)^2 \left (a+b \log \left (c x^n\right )\right )}{4 t}+\frac {b e n q (h s-g t)^2 \log (g+h x)}{4 h^2 t}-\frac {b e n s (g+h x) \log \left (f (g+h x)^q\right )}{2 h}-\frac {b n (s+t x)^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{4 t}-\frac {b n s^2 \log \left (-\frac {h x}{g}\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{2 t}+\frac {(s+t x)^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{2 t}-\frac {e q (h s-g t)^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {h x}{g}\right )}{2 h^2 t}-\frac {b e n q (h s-g t)^2 \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{2 h^2 t}-\frac {b e n q s^2 \operatorname {PolyLog}\left (2,1+\frac {h x}{g}\right )}{2 t} \] Output:
-1/2*b*d*n*s*x+3/4*b*e*n*q*s*x-1/2*a*e*q*(-g*t+h*s)*x/h+3/4*b*e*n*q*(-g*t+ h*s)*x/h+1/4*b*e*n*q*(t*x+s)^2/t+1/4*b*e*n*q*s^2*ln(x)/t-1/2*b*e*q*(-g*t+h *s)*x*ln(c*x^n)/h-1/4*e*q*(t*x+s)^2*(a+b*ln(c*x^n))/t+1/4*b*e*n*q*(-g*t+h* s)^2*ln(h*x+g)/h^2/t-1/2*b*e*n*s*(h*x+g)*ln(f*(h*x+g)^q)/h-1/4*b*n*(t*x+s) ^2*(d+e*ln(f*(h*x+g)^q))/t-1/2*b*n*s^2*ln(-h*x/g)*(d+e*ln(f*(h*x+g)^q))/t+ 1/2*(t*x+s)^2*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q))/t-1/2*e*q*(-g*t+h*s)^2 *(a+b*ln(c*x^n))*ln(1+h*x/g)/h^2/t-1/2*b*e*n*q*(-g*t+h*s)^2*polylog(2,-h*x /g)/h^2/t-1/2*b*e*n*q*s^2*polylog(2,1+h*x/g)/t
Time = 0.41 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.09 \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {4 a d h^2 s x-4 b d h^2 n s x-4 a e h^2 q s x+8 b e h^2 n q s x+2 a e g h q t x-3 b e g h n q t x+2 a d h^2 t x^2-b d h^2 n t x^2-a e h^2 q t x^2+b e h^2 n q t x^2+4 a e g h q s \log (g+h x)-4 b e g h n q s \log (g+h x)-2 a e g^2 q t \log (g+h x)+b e g^2 n q t \log (g+h x)-4 b e g h n q s \log (x) \log (g+h x)+2 b e g^2 n q t \log (x) \log (g+h x)+4 a e h^2 s x \log \left (f (g+h x)^q\right )-4 b e h^2 n s x \log \left (f (g+h x)^q\right )+2 a e h^2 t x^2 \log \left (f (g+h x)^q\right )-b e h^2 n t x^2 \log \left (f (g+h x)^q\right )+b \log \left (c x^n\right ) \left (-2 e g q (-2 h s+g t) \log (g+h x)+h x \left (2 d h (2 s+t x)-e q (4 h s-2 g t+h t x)+2 e h (2 s+t x) \log \left (f (g+h x)^q\right )\right )\right )+4 b e g h n q s \log (x) \log \left (1+\frac {h x}{g}\right )-2 b e g^2 n q t \log (x) \log \left (1+\frac {h x}{g}\right )-2 b e g n q (-2 h s+g t) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{4 h^2} \] Input:
Integrate[(s + t*x)*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]),x]
Output:
(4*a*d*h^2*s*x - 4*b*d*h^2*n*s*x - 4*a*e*h^2*q*s*x + 8*b*e*h^2*n*q*s*x + 2 *a*e*g*h*q*t*x - 3*b*e*g*h*n*q*t*x + 2*a*d*h^2*t*x^2 - b*d*h^2*n*t*x^2 - a *e*h^2*q*t*x^2 + b*e*h^2*n*q*t*x^2 + 4*a*e*g*h*q*s*Log[g + h*x] - 4*b*e*g* h*n*q*s*Log[g + h*x] - 2*a*e*g^2*q*t*Log[g + h*x] + b*e*g^2*n*q*t*Log[g + h*x] - 4*b*e*g*h*n*q*s*Log[x]*Log[g + h*x] + 2*b*e*g^2*n*q*t*Log[x]*Log[g + h*x] + 4*a*e*h^2*s*x*Log[f*(g + h*x)^q] - 4*b*e*h^2*n*s*x*Log[f*(g + h*x )^q] + 2*a*e*h^2*t*x^2*Log[f*(g + h*x)^q] - b*e*h^2*n*t*x^2*Log[f*(g + h*x )^q] + b*Log[c*x^n]*(-2*e*g*q*(-2*h*s + g*t)*Log[g + h*x] + h*x*(2*d*h*(2* s + t*x) - e*q*(4*h*s - 2*g*t + h*t*x) + 2*e*h*(2*s + t*x)*Log[f*(g + h*x) ^q])) + 4*b*e*g*h*n*q*s*Log[x]*Log[1 + (h*x)/g] - 2*b*e*g^2*n*q*t*Log[x]*L og[1 + (h*x)/g] - 2*b*e*g*n*q*(-2*h*s + g*t)*PolyLog[2, -((h*x)/g)])/(4*h^ 2)
Time = 1.82 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2890, 2889, 25, 27, 25, 2863, 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.
| \(\displaystyle \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2890 |
| \(\displaystyle \frac {\int (s+t x) \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )d(s+t x)}{t}\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle \frac {-\frac {e h q \int \frac {t (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{\left (g-\frac {h s}{t}\right ) t+h (s+t x)}d(s+t x)}{2 t}-\frac {b n \int \frac {(s+t x)^2 \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{x}d(s+t x)}{2 t}+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {e h q \int \frac {t (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{\left (g-\frac {h s}{t}\right ) t+h (s+t x)}d(s+t x)}{2 t}+\frac {b n \int -\frac {(s+t x)^2 \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{x}d(s+t x)}{2 t}+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{2} e h q \int -\frac {(s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{h s-g t-h (s+t x)}d(s+t x)+\frac {1}{2} b n \int -\frac {(s+t x)^2 \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t x}d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} e h q \int \frac {(s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{h s-g t-h (s+t x)}d(s+t x)+\frac {1}{2} b n \int -\frac {(s+t x)^2 \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t x}d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\frac {1}{2} e h q \int \left (\frac {\left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) (h s-g t)^2}{h^2 (h s-g t-h (s+t x))}+\frac {(g t-h s) \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{h^2}-\frac {(s+t x) \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{h}\right )d(s+t x)+\frac {1}{2} b n \int \left (-\frac {\left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right ) s^2}{t x}-\left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right ) s-(s+t x) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )\right )d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )}{t}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )+\frac {1}{2} e h q \left (-\frac {(h s-g t)^2 \log \left (-\frac {-g t-h (s+t x)+h s}{g t}\right ) \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{h^3}-\frac {(s+t x)^2 \left (a+b \log \left (c \left (\frac {s+t x}{t}-\frac {s}{t}\right )^n\right )\right )}{2 h}-\frac {a (s+t x) (h s-g t)}{h^2}-\frac {b t x \log \left (c x^n\right ) (h s-g t)}{h^2}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right ) (h s-g t)^2}{h^3}+\frac {b n (s+t x) (h s-g t)}{h^2}+\frac {b n s^2 \log (-t x)}{2 h}+\frac {b n s (s+t x)}{2 h}+\frac {b n (s+t x)^2}{4 h}\right )+\frac {1}{2} b n \left (s^2 \left (-\log \left (-\frac {h x}{g}\right )\right ) \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )-\frac {1}{2} (s+t x)^2 \left (d+e \log \left (f \left (g+\frac {h (s+t x)}{t}-\frac {h s}{t}\right )^q\right )\right )-d s (s+t x)+\frac {e s (-g t-h (s+t x)+h s) \log \left (f (g+h x)^q\right )}{h}+\frac {e q (h s-g t)^2 \log (-g t-h (s+t x)+h s)}{2 h^2}-e q s^2 \operatorname {PolyLog}\left (2,\frac {h x}{g}+1\right )+\frac {e q (s+t x) (h s-g t)}{2 h}+e q s (s+t x)+\frac {1}{4} e q (s+t x)^2\right )}{t}\) |
Input:
Int[(s + t*x)*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]),x]
Output:
(((s + t*x)^2*(a + b*Log[c*(-(s/t) + (s + t*x)/t)^n])*(d + e*Log[f*(g - (h *s)/t + (h*(s + t*x))/t)^q]))/2 + (e*h*q*((b*n*s*(s + t*x))/(2*h) - (a*(h* s - g*t)*(s + t*x))/h^2 + (b*n*(h*s - g*t)*(s + t*x))/h^2 + (b*n*(s + t*x) ^2)/(4*h) + (b*n*s^2*Log[-(t*x)])/(2*h) - (b*t*(h*s - g*t)*x*Log[c*x^n])/h ^2 - ((s + t*x)^2*(a + b*Log[c*(-(s/t) + (s + t*x)/t)^n]))/(2*h) - ((h*s - g*t)^2*Log[-((h*s - g*t - h*(s + t*x))/(g*t))]*(a + b*Log[c*(-(s/t) + (s + t*x)/t)^n]))/h^3 - (b*n*(h*s - g*t)^2*PolyLog[2, -((h*x)/g)])/h^3))/2 + (b*n*(-(d*s*(s + t*x)) + e*q*s*(s + t*x) + (e*q*(h*s - g*t)*(s + t*x))/(2* h) + (e*q*(s + t*x)^2)/4 + (e*s*(h*s - g*t - h*(s + t*x))*Log[f*(g + h*x)^ q])/h + (e*q*(h*s - g*t)^2*Log[h*s - g*t - h*(s + t*x)])/(2*h^2) - ((s + t *x)^2*(d + e*Log[f*(g - (h*s)/t + (h*(s + t*x))/t)^q]))/2 - s^2*Log[-((h*x )/g)]*(d + e*Log[f*(g - (h*s)/t + (h*(s + t*x))/t)^q]) - e*q*s^2*PolyLog[2 , 1 + (h*x)/g]))/2)/t
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.) *((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Simp[1/l Subst[Int[x^r*(a + b*Log[c*(-(e*k - d*l)/l + e*(x/l))^n])*(f + g*Log[h*(-(j*k - i*l)/l + j*(x/l))^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n}, x] && IntegerQ[r]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 107.38 (sec) , antiderivative size = 1496, normalized size of antiderivative = 3.74
Input:
int((t*x+s)*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q)),x,method=_RETURNVERBOSE)
Output:
1/4*I*q/h*e*x*Pi*b*g*t*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*q/h*e*x*Pi*b*g*t* csgn(I*c*x^n)^2*csgn(I*c)-1/4*I*q/h^2*e*g^2*ln(h*x+g)*Pi*b*t*csgn(I*x^n)*c sgn(I*c*x^n)^2+1/2*I*q/h*e*g*ln(h*x+g)*Pi*b*s*csgn(I*x^n)*csgn(I*c*x^n)^2- 1/4*I*q/h^2*e*g^2*ln(h*x+g)*Pi*b*t*csgn(I*c*x^n)^2*csgn(I*c)+1/2*I*q/h*e*g *ln(h*x+g)*Pi*b*s*csgn(I*c*x^n)^2*csgn(I*c)-1/4*q*a*t*e*x^2-1/2*q*a*e*g^2* t/h^2*ln(h*x+g)+1/2*g*q*a*t*e/h*x+1/4*q*b*e*n*t*x^2-a*e*q*s*x-5/8*q/h^2*b* e*n*g^2*t+q/h*b*e*n*g*s-1/4*q*ln(c)*b*e*t*x^2-q*ln(c)*b*e*s*x-1/4*q*b*e*ln (x^n)*t*x^2-q*b*e*ln(x^n)*x*s+(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*l n(f)*e+1/2*d)*((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)*(1/2*t*x^2+s*x)+b*ln(x^n)*x^2*t+2*b*ln(x^n)*x*s-1/2*b*n*t *x^2-2*x*s*b*n)+(1/2*b*e*x*(t*x+2*s)*ln(x^n)-1/4*I*e*(-Pi*b*t*x^2*csgn(I*x ^n)*csgn(I*c*x^n)^2+Pi*b*t*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+Pi*b*t* x^2*csgn(I*c*x^n)^3-Pi*b*t*x^2*csgn(I*c*x^n)^2*csgn(I*c)-2*x*Pi*b*s*csgn(I *x^n)*csgn(I*c*x^n)^2+2*x*Pi*b*s*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2*x*P i*b*s*csgn(I*c*x^n)^3-2*x*Pi*b*s*csgn(I*c*x^n)^2*csgn(I*c)+2*I*ln(c)*b*t*x ^2-I*b*n*t*x^2+4*I*x*ln(c)*b*s+2*I*a*t*x^2-4*I*x*s*b*n+4*I*x*a*s))*ln((h*x +g)^q)-1/4*I*q/h*e*x*Pi*b*g*t*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/4*I...
\[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int { {\left (t x + s\right )} {\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate((t*x+s)*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="fr icas")
Output:
integral(a*d*t*x + a*d*s + (a*e*t*x + a*e*s + (b*e*t*x + b*e*s)*log(c*x^n) )*log((h*x + g)^q*f) + (b*d*t*x + b*d*s)*log(c*x^n), x)
Timed out. \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((t*x+s)*(a+b*ln(c*x**n))*(d+e*ln(f*(h*x+g)**q)),x)
Output:
Timed out
Time = 0.35 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.35 \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=-\frac {1}{4} \, b d n t x^{2} - a e h q s {\left (\frac {x}{h} - \frac {g \log \left (h x + g\right )}{h^{2}}\right )} - \frac {1}{4} \, a e h q t {\left (\frac {2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac {h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac {1}{2} \, a e t x^{2} \log \left ({\left (h x + g\right )}^{q} f\right ) + \frac {1}{2} \, b d t x^{2} \log \left (c x^{n}\right ) - b d n s x + \frac {1}{2} \, a d t x^{2} + a e s x \log \left ({\left (h x + g\right )}^{q} f\right ) + b d s x \log \left (c x^{n}\right ) + a d s x + \frac {{\left (2 \, g h n q s - g^{2} n q t\right )} {\left (\log \left (\frac {h x}{g} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {h x}{g}\right )\right )} b e}{2 \, h^{2}} + \frac {{\left (2 \, {\left (2 \, g h q s - g^{2} q t\right )} e \log \left (c\right ) - {\left (4 \, g h n q s - g^{2} n q t\right )} e\right )} b \log \left (h x + g\right )}{4 \, h^{2}} - \frac {2 \, {\left (2 \, g h n q s - g^{2} n q t\right )} b e \log \left (h x + g\right ) \log \left (x\right ) + {\left ({\left (h^{2} q t - 2 \, h^{2} t \log \left (f\right )\right )} e \log \left (c\right ) - {\left (h^{2} n q t - h^{2} n t \log \left (f\right )\right )} e\right )} b x^{2} + {\left (2 \, {\left (2 \, h^{2} q s - g h q t - 2 \, h^{2} s \log \left (f\right )\right )} e \log \left (c\right ) - {\left (8 \, h^{2} n q s - 3 \, g h n q t - 4 \, h^{2} n s \log \left (f\right )\right )} e\right )} b x + {\left ({\left (e h^{2} n t - 2 \, e h^{2} t \log \left (c\right )\right )} b x^{2} + 4 \, {\left (e h^{2} n s - e h^{2} s \log \left (c\right )\right )} b x - 2 \, {\left (b e h^{2} t x^{2} + 2 \, b e h^{2} s x\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (h x + g\right )}^{q}\right ) + {\left ({\left (h^{2} q t - 2 \, h^{2} t \log \left (f\right )\right )} b e x^{2} + 2 \, {\left (2 \, h^{2} q s - g h q t - 2 \, h^{2} s \log \left (f\right )\right )} b e x - 2 \, {\left (2 \, g h q s - g^{2} q t\right )} b e \log \left (h x + g\right )\right )} \log \left (x^{n}\right )}{4 \, h^{2}} \] Input:
integrate((t*x+s)*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="ma xima")
Output:
-1/4*b*d*n*t*x^2 - a*e*h*q*s*(x/h - g*log(h*x + g)/h^2) - 1/4*a*e*h*q*t*(2 *g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + 1/2*a*e*t*x^2*log((h*x + g) ^q*f) + 1/2*b*d*t*x^2*log(c*x^n) - b*d*n*s*x + 1/2*a*d*t*x^2 + a*e*s*x*log ((h*x + g)^q*f) + b*d*s*x*log(c*x^n) + a*d*s*x + 1/2*(2*g*h*n*q*s - g^2*n* q*t)*(log(h*x/g + 1)*log(x) + dilog(-h*x/g))*b*e/h^2 + 1/4*(2*(2*g*h*q*s - g^2*q*t)*e*log(c) - (4*g*h*n*q*s - g^2*n*q*t)*e)*b*log(h*x + g)/h^2 - 1/4 *(2*(2*g*h*n*q*s - g^2*n*q*t)*b*e*log(h*x + g)*log(x) + ((h^2*q*t - 2*h^2* t*log(f))*e*log(c) - (h^2*n*q*t - h^2*n*t*log(f))*e)*b*x^2 + (2*(2*h^2*q*s - g*h*q*t - 2*h^2*s*log(f))*e*log(c) - (8*h^2*n*q*s - 3*g*h*n*q*t - 4*h^2 *n*s*log(f))*e)*b*x + ((e*h^2*n*t - 2*e*h^2*t*log(c))*b*x^2 + 4*(e*h^2*n*s - e*h^2*s*log(c))*b*x - 2*(b*e*h^2*t*x^2 + 2*b*e*h^2*s*x)*log(x^n))*log(( h*x + g)^q) + ((h^2*q*t - 2*h^2*t*log(f))*b*e*x^2 + 2*(2*h^2*q*s - g*h*q*t - 2*h^2*s*log(f))*b*e*x - 2*(2*g*h*q*s - g^2*q*t)*b*e*log(h*x + g))*log(x ^n))/h^2
\[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int { {\left (t x + s\right )} {\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate((t*x+s)*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="gi ac")
Output:
integrate((t*x + s)*(e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a), x)
Timed out. \[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\int \left (s+t\,x\right )\,\left (d+e\,\ln \left (f\,{\left (g+h\,x\right )}^q\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:
int((s + t*x)*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)),x)
Output:
int((s + t*x)*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)), x)
\[ \int (s+t x) \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {-\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,h^{2} n^{2} t \,x^{2}-\mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n q t \,x^{2}-a e \,h^{2} n q t \,x^{2}+4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n s x +2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n t \,x^{2}-4 \,\mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n q s x +2 a e g h n q t x -3 b e g h \,n^{2} q t x +\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,g^{2} n^{2} t -\mathrm {log}\left (x^{n} c \right )^{2} b e \,g^{2} q t -b d \,h^{2} n^{2} t \,x^{2}-2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,g^{2} n t +4 a d \,h^{2} n s x +2 a d \,h^{2} n t \,x^{2}-4 b d \,h^{2} n^{2} s x +b e \,h^{2} n^{2} q t \,x^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) b e g h n q t x +4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e g h n s +4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,h^{2} n s x +2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,h^{2} n t \,x^{2}-4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e g h \,n^{2} s -4 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,h^{2} n^{2} s x +2 \mathrm {log}\left (x^{n} c \right )^{2} b e g h q s +4 \,\mathrm {log}\left (x^{n} c \right ) b d \,h^{2} n s x +2 \,\mathrm {log}\left (x^{n} c \right ) b d \,h^{2} n t \,x^{2}-4 a e \,h^{2} n q s x +8 b e \,h^{2} n^{2} q s x +2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{2}+g x}d x \right ) b e \,g^{3} n q t -4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{2}+g x}d x \right ) b e \,g^{2} h n q s}{4 h^{2} n} \] Input:
int((t*x+s)*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x)
Output:
(2*int(log(x**n*c)/(g*x + h*x**2),x)*b*e*g**3*n*q*t - 4*int(log(x**n*c)/(g *x + h*x**2),x)*b*e*g**2*h*n*q*s + 4*log((g + h*x)**q*f)*log(x**n*c)*b*e*h **2*n*s*x + 2*log((g + h*x)**q*f)*log(x**n*c)*b*e*h**2*n*t*x**2 - 2*log((g + h*x)**q*f)*a*e*g**2*n*t + 4*log((g + h*x)**q*f)*a*e*g*h*n*s + 4*log((g + h*x)**q*f)*a*e*h**2*n*s*x + 2*log((g + h*x)**q*f)*a*e*h**2*n*t*x**2 + lo g((g + h*x)**q*f)*b*e*g**2*n**2*t - 4*log((g + h*x)**q*f)*b*e*g*h*n**2*s - 4*log((g + h*x)**q*f)*b*e*h**2*n**2*s*x - log((g + h*x)**q*f)*b*e*h**2*n* *2*t*x**2 - log(x**n*c)**2*b*e*g**2*q*t + 2*log(x**n*c)**2*b*e*g*h*q*s + 4 *log(x**n*c)*b*d*h**2*n*s*x + 2*log(x**n*c)*b*d*h**2*n*t*x**2 + 2*log(x**n *c)*b*e*g*h*n*q*t*x - 4*log(x**n*c)*b*e*h**2*n*q*s*x - log(x**n*c)*b*e*h** 2*n*q*t*x**2 + 4*a*d*h**2*n*s*x + 2*a*d*h**2*n*t*x**2 + 2*a*e*g*h*n*q*t*x - 4*a*e*h**2*n*q*s*x - a*e*h**2*n*q*t*x**2 - 4*b*d*h**2*n**2*s*x - b*d*h** 2*n**2*t*x**2 - 3*b*e*g*h*n**2*q*t*x + 8*b*e*h**2*n**2*q*s*x + b*e*h**2*n* *2*q*t*x**2)/(4*h**2*n)