Integrand size = 45, antiderivative size = 392 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {2 A B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}+\frac {2 B^2 d^2 n^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B^2 n^2 (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g^2 i^2 n} \]
-2*A*B*d^2*n*(b*x+a)/(-a*d+b*c)^3/g^2/i^2/(d*x+c)+2*B^2*d^2*n^2*(b*x+a)/(- a*d+b*c)^3/g^2/i^2/(d*x+c)-2*b^2*B^2*n^2*(d*x+c)/(-a*d+b*c)^3/g^2/i^2/(b*x +a)-2*B^2*d^2*n*(b*x+a)*ln(e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)^3/g^2/i^2/(d* x+c)-2*b^2*B*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^2/i^ 2/(b*x+a)+d^2*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^2/i ^2/(d*x+c)-b^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^2/ i^2/(b*x+a)-2/3*b*d*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^3/g^2/i ^2/n
Leaf count is larger than twice the leaf count of optimal. \(870\) vs. \(2(392)=784\).
Time = 0.62 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.22 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {2 b B^2 d n^2 (a+b x) (c+d x) \log ^3\left (\frac {a+b x}{c+d x}\right )+3 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (2 a A b c d+b^2 B c^2 n-a^2 B d^2 n+2 A b^2 c d x+2 a A b d^2 x+2 b^2 B c d n x-2 a b B d^2 n x+2 A b^2 d^2 x^2+2 b B d (a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 b B d n (a+b x) (c+d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (A b c+a A d+b B c n-a B d n+2 A b d x+B (a d+b (c+2 d x)) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n (b c+a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 b d (a+b x) (c+d x) \log (a+b x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right )+3 b (b c-a d) (c+d x) \left (A^2+2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n (A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (A+B n-B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+3 d (b c-a d) (a+b x) \left (A^2-2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-A+B n+B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b d (a+b x) (c+d x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right ) \log (c+d x)}{3 (b c-a d)^3 g^2 i^2 (a+b x) (c+d x)} \]
-1/3*(2*b*B^2*d*n^2*(a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^3 + 3*B*n *Log[(a + b*x)/(c + d*x)]^2*(2*a*A*b*c*d + b^2*B*c^2*n - a^2*B*d^2*n + 2*A *b^2*c*d*x + 2*a*A*b*d^2*x + 2*b^2*B*c*d*n*x - 2*a*b*B*d^2*n*x + 2*A*b^2*d ^2*x^2 + 2*b*B*d*(a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*b* B*d*n*(a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]) + 6*B*(b*c - a*d)*n*Lo g[(a + b*x)/(c + d*x)]*(A*b*c + a*A*d + b*B*c*n - a*B*d*n + 2*A*b*d*x + B* (a*d + b*(c + 2*d*x))*Log[e*((a + b*x)/(c + d*x))^n] - B*n*(b*c + a*d + 2* b*d*x)*Log[(a + b*x)/(c + d*x)]) + 6*b*d*(a + b*x)*(c + d*x)*Log[a + b*x]* (A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x) /(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2) + 3*b*(b*c - a*d)*(c + d*x)*(A^2 + 2*A*B*n + 2*B^2*n^2 + B^2*Log [e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(A + B*n)*Log[(a + b*x)/(c + d*x)] + B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*( A + B*n - B*n*Log[(a + b*x)/(c + d*x)])) + 3*d*(b*c - a*d)*(a + b*x)*(A^2 - 2*A*B*n + 2*B^2*n^2 + B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-A + B*n)*Log[(a + b*x)/(c + d*x)] + B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B* Log[e*((a + b*x)/(c + d*x))^n]*(-A + B*n + B*n*Log[(a + b*x)/(c + d*x)])) - 6*b*d*(a + b*x)*(c + d*x)*(A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x) )^n] - n*Log[(a + b*x)/(c + d*x)])^2)*Log[c + d*x])/((b*c - a*d)^3*g^2*...
Time = 0.55 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 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.
| \(\displaystyle \int \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 i^2 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2}-\frac {2 b d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}\right )d\frac {a+b x}{c+d x}}{g^2 i^2 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {2 b^2 B n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {2 A B d^2 n (a+b x)}{c+d x}-\frac {2 b d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B n}-\frac {2 b^2 B^2 n^2 (c+d x)}{a+b x}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 B^2 d^2 n^2 (a+b x)}{c+d x}}{g^2 i^2 (b c-a d)^3}\) |
((-2*A*B*d^2*n*(a + b*x))/(c + d*x) + (2*B^2*d^2*n^2*(a + b*x))/(c + d*x) - (2*b^2*B^2*n^2*(c + d*x))/(a + b*x) - (2*B^2*d^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(c + d*x) - (2*b^2*B*n*(c + d*x)*(A + B*Log[e*((a + b *x)/(c + d*x))^n]))/(a + b*x) + (d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(c + d*x) - (b^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x ))^n])^2)/(a + b*x) - (2*b*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3* B*n))/((b*c - a*d)^3*g^2*i^2)
3.2.99.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(1306\) vs. \(2(390)=780\).
Time = 9.72 (sec) , antiderivative size = 1307, normalized size of antiderivative = 3.33
1/3*(-12*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^3*d^2*n+12*A*B*x*ln(e*((b *x+a)/(d*x+c))^n)*a^3*b^2*c^4*d*n+12*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3 *b^2*c^3*d^2*n^2+6*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b^2*c^3*d^2-6*A *B*x^2*a^4*b*c^2*d^3*n^2+12*A*B*x^2*a^3*b^2*c^3*d^2*n^2-6*A*B*x^2*a^2*b^3* c^4*d*n^2-6*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b*c^3*d^2*n+6*B^2*x*ln(e *((b*x+a)/(d*x+c))^n)^2*a^3*b^2*c^4*d*n+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n) *a^4*b*c^3*d^2*n^2+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^4*d*n^2+6* A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b*c^3*d^2+6*A*B*x*ln(e*((b*x+a)/(d*x +c))^n)^2*a^3*b^2*c^4*d+6*A*B*x*a^4*b*c^3*d^2*n^2+6*A*B*x*a^3*b^2*c^4*d*n^ 2+6*B^2*x^2*a^4*b*c^2*d^3*n^3-6*B^2*x^2*a^2*b^3*c^4*d*n^3+2*B^2*x*ln(e*((b *x+a)/(d*x+c))^n)^3*a^4*b*c^3*d^2+2*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^3* b^2*c^4*d-6*B^2*x*a^4*b*c^3*d^2*n^3+6*B^2*x*a^3*b^2*c^4*d*n^3+6*A^2*x^2*ln (e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^3*d^2+3*A^2*x^2*a^4*b*c^2*d^3*n-3*A^2*x^ 2*a^2*b^3*c^4*d*n-6*A*B*x*a^5*c^2*d^3*n^2-6*A*B*x*a^2*b^3*c^5*n^2+6*A^2*x* ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^3*d^2+6*A^2*x*ln(e*((b*x+a)/(d*x+c))^n)* a^3*b^2*c^4*d-3*A^2*x*a^4*b*c^3*d^2*n+3*A^2*x*a^3*b^2*c^4*d*n+6*A*B*ln(e*( (b*x+a)/(d*x+c))^n)^2*a^4*b*c^4*d-6*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*c^3* d^2*n+6*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^5*n+3*B^2*ln(e*((b*x+a)/(d *x+c))^n)^2*a^3*b^2*c^5*n+6*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^5*c^3*d^2*n^2+ 6*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^5*n^2+3*A^2*x*a^5*c^2*d^3*n-3...
Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (390) = 780\).
Time = 0.38 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.51 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {3 \, A^{2} b^{2} c^{2} - 3 \, A^{2} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + B^{2} a b c d n^{2} + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 6 \, {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 3 \, {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + B^{2} a b c d + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right )^{2} + 3 \, {\left (2 \, A B b^{2} d^{2} n x^{2} + 2 \, A B a b c d n + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + {\left (A B b^{2} c d + A B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, {\left (A B b^{2} c^{2} - 2 \, A B a b c d + A B a^{2} d^{2}\right )} n + 6 \, {\left (A^{2} b^{2} c d - A^{2} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2}\right )} x + 6 \, {\left (A B b^{2} c^{2} - A B a^{2} d^{2} + {\left (B^{2} b^{2} d^{2} n x^{2} + B^{2} a b c d n + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} x + {\left (2 \, A B b^{2} d^{2} x^{2} + 2 \, A B a b c d + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (A B b^{2} c d + A B a b d^{2} + {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 6 \, {\left (A^{2} a b c d + {\left (B^{2} b^{2} c^{2} + B^{2} a^{2} d^{2}\right )} n^{2} + {\left (2 \, B^{2} b^{2} d^{2} n^{2} + A^{2} b^{2} d^{2}\right )} x^{2} + {\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} n + {\left (A^{2} b^{2} c d + A^{2} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}\right )}} \]
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x , algorithm="fricas")
-1/3*(3*A^2*b^2*c^2 - 3*A^2*a^2*d^2 + 2*(B^2*b^2*d^2*n^2*x^2 + B^2*a*b*c*d *n^2 + (B^2*b^2*c*d + B^2*a*b*d^2)*n^2*x)*log((b*x + a)/(d*x + c))^3 + 6*( B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 3*(B^2*b^2*c^2 - B^2*a^2*d^2 + 2*(B^2*b^2 *c*d - B^2*a*b*d^2)*x + 2*(B^2*b^2*d^2*x^2 + B^2*a*b*c*d + (B^2*b^2*c*d + B^2*a*b*d^2)*x)*log((b*x + a)/(d*x + c)))*log(e)^2 + 3*(2*A*B*b^2*d^2*n*x^ 2 + 2*A*B*a*b*c*d*n + (B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 2*((B^2*b^2*c*d - B^2*a*b*d^2)*n^2 + (A*B*b^2*c*d + A*B*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c))^2 + 6*(A*B*b^2*c^2 - 2*A*B*a*b*c*d + A*B*a^2*d^2)*n + 6*(A^2*b^2*c*d - A^2*a*b*d^2 + 2*(B^2*b^2*c*d - B^2*a*b*d^2)*n^2)*x + 6*(A*B*b^2*c^2 - A*B *a^2*d^2 + (B^2*b^2*d^2*n*x^2 + B^2*a*b*c*d*n + (B^2*b^2*c*d + B^2*a*b*d^2 )*n*x)*log((b*x + a)/(d*x + c))^2 + (B^2*b^2*c^2 - 2*B^2*a*b*c*d + B^2*a^2 *d^2)*n + 2*(A*B*b^2*c*d - A*B*a*b*d^2)*x + (2*A*B*b^2*d^2*x^2 + 2*A*B*a*b *c*d + (B^2*b^2*c^2 - B^2*a^2*d^2)*n + 2*(A*B*b^2*c*d + A*B*a*b*d^2 + (B^2 *b^2*c*d - B^2*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) + 6*(A^2*a* b*c*d + (B^2*b^2*c^2 + B^2*a^2*d^2)*n^2 + (2*B^2*b^2*d^2*n^2 + A^2*b^2*d^2 )*x^2 + (A*B*b^2*c^2 - A*B*a^2*d^2)*n + (A^2*b^2*c*d + A^2*a*b*d^2 + 2*(B^ 2*b^2*c*d + B^2*a*b*d^2)*n^2 + 2*(A*B*b^2*c*d - A*B*a*b*d^2)*n)*x)*log((b* x + a)/(d*x + c)))/((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b *d^4)*g^2*i^2*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*g^ 2*i^2*x + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*g...
Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (390) = 780\).
Time = 0.31 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.12 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Too large to display} \]
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x , algorithm="maxima")
-B^2*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*g^2*i ^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*g^2*i^2*x + (a*b^ 2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2) - 2*b*d*log(d*x + c)/ ((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2))*log(e*(b*x/ (d*x + c) + a/(d*x + c))^n)^2 - 2*A*B*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*g^2*i^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c *d^2 + a^3*d^3)*g^2*i^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^ 2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^ 3)*g^2*i^2) - 2*b*d*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 2/3*((3*b^2 *c^2 - 3*a^2*d^2 + (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^3 + 3*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*l og(d*x + c)^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(d*x + c)^3 + 6*(b^2*c*d - a*b*d^2)*x + 6*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b *d^2)*x)*log(b*x + a) - 3*(2*b^2*d^2*x^2 + 2*a*b*c*d + (b^2*d^2*x^2 + a*b* c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^2 + 2*(b^2*c*d + a*b*d^2)*x)*log (d*x + c))*n^2/(a*b^3*c^4*g^2*i^2 - 3*a^2*b^2*c^3*d*g^2*i^2 + 3*a^3*b*c^2* d^2*g^2*i^2 - a^4*c*d^3*g^2*i^2 + (b^4*c^3*d*g^2*i^2 - 3*a*b^3*c^2*d^2*g^2 *i^2 + 3*a^2*b^2*c*d^3*g^2*i^2 - a^3*b*d^4*g^2*i^2)*x^2 + (b^4*c^4*g^2*...
Time = 285.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.47 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-{\left (\frac {{\left (d x + c\right )} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )} g^{2} i^{2}} + \frac {2 \, {\left (B^{2} n^{2} + B^{2} n \log \left (e\right ) + A B n\right )} {\left (d x + c\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2} i^{2}} + \frac {{\left (2 \, B^{2} n^{2} + 2 \, B^{2} n \log \left (e\right ) + B^{2} \log \left (e\right )^{2} + 2 \, A B n + 2 \, A B \log \left (e\right ) + A^{2}\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2} i^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x , algorithm="giac")
-((d*x + c)*B^2*n^2*log((b*x + a)/(d*x + c))^2/((b*x + a)*g^2*i^2) + 2*(B^ 2*n^2 + B^2*n*log(e) + A*B*n)*(d*x + c)*log((b*x + a)/(d*x + c))/((b*x + a )*g^2*i^2) + (2*B^2*n^2 + 2*B^2*n*log(e) + B^2*log(e)^2 + 2*A*B*n + 2*A*B* log(e) + A^2)*(d*x + c)/((b*x + a)*g^2*i^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)^2
Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^2} \,d x \]