Integrand size = 33, antiderivative size = 570 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 h^2 n^2 x}{3 b^2 d^2}+\frac {B^2 (b c-a d)^3 h^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 h^2 n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^2 \log (c+d x)}{3 b^3 d^3}-\frac {2 B (b c-a d) h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b d^3}+\frac {2 B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b^3 d^3}-\frac {(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 b^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 h}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3} \]
1/3*B^2*(-a*d+b*c)^2*h^2*n^2*x/b^2/d^2+1/3*B^2*(-a*d+b*c)^3*h^2*n^2*ln((b* x+a)/(d*x+c))/b^3/d^3+1/3*B^2*(-a*d+b*c)^3*h^2*n^2*ln(d*x+c)/b^3/d^3+2/3*B ^2*(-a*d+b*c)^2*h*(-a*d*h-2*b*c*h+3*b*d*g)*n^2*ln(d*x+c)/b^3/d^3-2/3*B*(-a *d+b*c)*h*(-a*d*h-2*b*c*h+3*b*d*g)*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^ n)))/b^3/d^2-1/3*B*(-a*d+b*c)*h^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c) ^n)))/b/d^3+2/3*B*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^ 2-3*c*d*g*h+3*d^2*g^2))*n*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d *x+c)^n)))/b^3/d^3-1/3*(-a*h+b*g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b^ 3/h+1/3*(h*x+g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/h+2/3*B^2*(-a*d+b*c) *(a^2*d^2*h^2-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n^2* polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3
Time = 1.07 (sec) , antiderivative size = 906, normalized size of antiderivative = 1.59 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\frac {-a B^2 d^3 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) n^2 \log ^2(a+b x)+B n \log (a+b x) \left (2 b^3 B c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) n \log (c+d x)+2 B \left (3 a b^2 d^3 g^2-3 a^2 b d^3 g h+a^3 d^3 h^2-b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (2 A d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right )+B \left (-3 a^2 d^2 h^2+a b d h (6 d g+c h)+2 b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n+2 B d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+b \left (-b^2 B^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) n^2 \log ^2(c+d x)+B n \log (c+d x) \left (-2 A b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )+B \left (2 a^2 c d^2 h^2-3 b^2 c^2 h (-2 d g+c h)+a b d \left (-6 d^2 g^2-6 c d g h+c^2 h^2\right )\right ) n-2 b^2 B c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+d \left (a^2 B d^2 h^2 n (-2 A+B n) x+a b B n \left (A d^2 \left (-6 g^2+6 g h x+h^2 x^2\right )-2 B n \left (3 d^2 g^2+c^2 h^2+c d h (-3 g+h x)\right )\right )+b^2 x \left (B^2 c^2 h^2 n^2+A^2 d^2 \left (3 g^2+3 g h x+h^2 x^2\right )+A B c h n (2 c h-d (6 g+h x))\right )+B \left (-2 a^2 B d^2 h^2 n x+a b B d^2 n \left (-6 g^2+6 g h x+h^2 x^2\right )+b^2 x \left (B c h n (-6 d g+2 c h-d h x)+2 A d^2 \left (3 g^2+3 g h x+h^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+b^2 B^2 d^2 x \left (3 g^2+3 g h x+h^2 x^2\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+2 B^2 \left (3 a b^2 d^3 g^2-3 a^2 b d^3 g h+a^3 d^3 h^2-b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{3 b^3 d^3} \]
(-(a*B^2*d^3*(3*b^2*g^2 - 3*a*b*g*h + a^2*h^2)*n^2*Log[a + b*x]^2) + B*n*L og[a + b*x]*(2*b^3*B*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2)*n*Log[c + d*x] + 2*B*(3*a*b^2*d^3*g^2 - 3*a^2*b*d^3*g*h + a^3*d^3*h^2 - b^3*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(2*A*d^2*(3*b ^2*g^2 - 3*a*b*g*h + a^2*h^2) + B*(-3*a^2*d^2*h^2 + a*b*d*h*(6*d*g + c*h) + 2*b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n + 2*B*d^2*(3*b^2*g^2 - 3*a*b* g*h + a^2*h^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n])) + b*(-(b^2*B^2*c*(3*d^2* g^2 - 3*c*d*g*h + c^2*h^2)*n^2*Log[c + d*x]^2) + B*n*Log[c + d*x]*(-2*A*b^ 2*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2) + B*(2*a^2*c*d^2*h^2 - 3*b^2*c^2*h*( -2*d*g + c*h) + a*b*d*(-6*d^2*g^2 - 6*c*d*g*h + c^2*h^2))*n - 2*b^2*B*c*(3 *d^2*g^2 - 3*c*d*g*h + c^2*h^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + d*(a^2 *B*d^2*h^2*n*(-2*A + B*n)*x + a*b*B*n*(A*d^2*(-6*g^2 + 6*g*h*x + h^2*x^2) - 2*B*n*(3*d^2*g^2 + c^2*h^2 + c*d*h*(-3*g + h*x))) + b^2*x*(B^2*c^2*h^2*n ^2 + A^2*d^2*(3*g^2 + 3*g*h*x + h^2*x^2) + A*B*c*h*n*(2*c*h - d*(6*g + h*x ))) + B*(-2*a^2*B*d^2*h^2*n*x + a*b*B*d^2*n*(-6*g^2 + 6*g*h*x + h^2*x^2) + b^2*x*(B*c*h*n*(-6*d*g + 2*c*h - d*h*x) + 2*A*d^2*(3*g^2 + 3*g*h*x + h^2* x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + b^2*B^2*d^2*x*(3*g^2 + 3*g*h*x + h^2*x^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)) + 2*B^2*(3*a*b^2*d^3*g^2 - 3*a^2*b*d^3*g*h + a^3*d^3*h^2 - b^3*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n ^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b^3*d^3)
Time = 1.21 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2953, 2798, 2804, 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.
| \(\displaystyle \int (g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int (g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2798 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \frac {(c+d x) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \left (\frac {(b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) h^3}{b d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {(b c-a d)^2 (3 b d g-2 b c h-a d h) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) h^2}{b^2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) h}{b^3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b g-a h)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{3 h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (-\frac {h (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 d^3}-\frac {B h n (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}+\frac {h^2 (a+b x) (b c-a d)^2 (-a d h-2 b c h+3 b d g) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 d^2 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b g-a h)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^3 B n}+\frac {h^3 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {B h^2 n (b c-a d)^2 (-a d h-2 b c h+3 b d g) \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3 d^3}-\frac {B h^3 n (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 d^3}+\frac {B h^3 n (b c-a d)^3 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{2 b^3 d^3}-\frac {B h^3 n (b c-a d)^3}{2 b^2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{3 h (b c-a d)}\right )\) |
(b*c - a*d)*(((b*g - a*h - ((d*g - c*h)*(a + b*x))/(c + d*x))^3*(A + B*Log [e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)*h*(b - (d*(a + b*x))/(c + d *x))^3) - (2*B*n*(-1/2*(B*(b*c - a*d)^3*h^3*n)/(b^2*d^3*(b - (d*(a + b*x)) /(c + d*x))) + ((b*c - a*d)^3*h^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/ (2*b*d^3*(b - (d*(a + b*x))/(c + d*x))^2) + ((b*c - a*d)^2*h^2*(3*b*d*g - 2*b*c*h - a*d*h)*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*d^ 2*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + ((b*g - a*h)^3*(A + B*Log[e*( (a + b*x)/(c + d*x))^n])^2)/(2*b^3*B*n) - (B*(b*c - a*d)^3*h^3*n*Log[(a + b*x)/(c + d*x)])/(2*b^3*d^3) + (B*(b*c - a*d)^3*h^3*n*Log[b - (d*(a + b*x) )/(c + d*x)])/(2*b^3*d^3) + (B*(b*c - a*d)^2*h^2*(3*b*d*g - 2*b*c*h - a*d* h)*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b^3*d^3) - ((b*c - a*d)*h*(a^2*d^2 *h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/( b^3*d^3) - (B*(b*c - a*d)*h*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3* d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))]) /(b^3*d^3)))/(3*(b*c - a*d)*h))
3.4.3.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) *(e*f - d*g))) Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Sub st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 )), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} , x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 90.91 (sec) , antiderivative size = 8443, normalized size of antiderivative = 14.81
\[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\int { {\left (h x + g\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2} \,d x } \]
integral(A^2*h^2*x^2 + 2*A^2*g*h*x + A^2*g^2 + (B^2*h^2*x^2 + 2*B^2*g*h*x + B^2*g^2)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 2*(A*B*h^2*x^2 + 2*A*B*g*h*x + A*B*g^2)*log((b*x + a)^n*e/(d*x + c)^n), x)
Exception generated. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Leaf count of result is larger than twice the leaf count of optimal. 1671 vs. \(2 (549) = 1098\).
Time = 0.76 (sec) , antiderivative size = 1671, normalized size of antiderivative = 2.93 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Too large to display} \]
2/3*A*B*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^2*h^2*x^3 + 2*A*B*g *h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + A^2*g*h*x^2 + 2*A*B*g^2*x*log((b*x + a)^n*e/(d*x + c)^n) + A^2*g^2*x + 2*(a*e*n*log(b*x + a)/b - c*e*n*log(d *x + c)/d)*A*B*g^2/e - 2*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/ d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A*B*g*h/e + 1/3*(2*a^3*e*n*log(b*x + a) /b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 - 2*( b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A*B*h^2/e + 1/3*(2*a^2*c*d^2*h^2* n^2 - (6*c*d^2*g*h*n^2 - c^2*d*h^2*n^2)*a*b - (6*c*d^2*g^2*n*log(e) + (3*h ^2*n^2 + 2*h^2*n*log(e))*c^3 - 6*(g*h*n^2 + g*h*n*log(e))*c^2*d)*b^2)*B^2* log(d*x + c)/(b^2*d^3) + 2/3*(3*a*b^2*d^3*g^2*n^2 - 3*a^2*b*d^3*g*h*n^2 + a^3*d^3*h^2*n^2 - (3*c*d^2*g^2*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*b^3)*( log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b* c - a*d)))*B^2/(b^3*d^3) + 1/3*(B^2*b^3*d^3*h^2*x^3*log(e)^2 + 2*(3*c*d^2* g^2*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*B^2*b^3*log(b*x + a)*log(d*x + c) - (3*c*d^2*g^2*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*B^2*b^3*log(d*x + c)^ 2 + (a*b^2*d^3*h^2*n*log(e) - (c*d^2*h^2*n*log(e) - 3*d^3*g*h*log(e)^2)*b^ 3)*B^2*x^2 - (3*a*b^2*d^3*g^2*n^2 - 3*a^2*b*d^3*g*h*n^2 + a^3*d^3*h^2*n^2) *B^2*log(b*x + a)^2 + ((h^2*n^2 - 2*h^2*n*log(e))*a^2*b*d^3 - 2*(c*d^2*h^2 *n^2 - 3*d^3*g*h*n*log(e))*a*b^2 - (6*c*d^2*g*h*n*log(e) - 3*d^3*g^2*log(e )^2 - (h^2*n^2 + 2*h^2*n*log(e))*c^2*d)*b^3)*B^2*x - ((3*h^2*n^2 - 2*h^...
Timed out. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Timed out} \]
Timed out. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\int {\left (g+h\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2 \,d x \]