Integrand size = 45, antiderivative size = 231 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {2 A B d n (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}-\frac {2 B^2 d n^2 (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}+\frac {2 B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^2 g i^2 (c+d x)}-\frac {d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^2 g i^2 (c+d x)}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^2 g i^2 n} \]
2*A*B*d*n*(b*x+a)/(-a*d+b*c)^2/g/i^2/(d*x+c)-2*B^2*d*n^2*(b*x+a)/(-a*d+b*c )^2/g/i^2/(d*x+c)+2*B^2*d*n*(b*x+a)*ln(e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)^2 /g/i^2/(d*x+c)-d*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g/ i^2/(d*x+c)+1/3*b*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^2/g/i^2/n
Leaf count is larger than twice the leaf count of optimal. \(789\) vs. \(2(231)=462\).
Time = 0.40 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.42 \[ \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) (c i+d i x)^2} \, dx=\frac {b B^2 n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^2 g i^2}-\frac {2 B n \log \left (\frac {a+b x}{c+d x}\right ) \left (-A+B n-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 )\right )}{(b c-a d) g i^2 (c+d x)}+\frac {\log ^2\left (\frac {a+b x}{c+d x}\right ) \left (A b B c n-a B^2 d n^2+A b B d n x-b B^2 d n^2 x+b B^2 c n \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 B^2 d n x \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 )\right )}{(b c-a d)^2 g i^2 (c+d x)}+\frac {A^2-2 A B n+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 )-2 B^2 n \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}{(b c-a d) g i^2 (c+d x)}+\frac {b \log (a+b x) \left (A^2-2 A B n+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 )-2 B^2 n \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 )}{(b c-a d)^2 g i^2}-\frac {b \left (A^2-2 A B n+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 )-2 B^2 n \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)}{(b c-a d)^2 g i^2} \]
(b*B^2*n^2*Log[(a + b*x)/(c + d*x)]^3)/(3*(b*c - a*d)^2*g*i^2) - (2*B*n*Lo g[(a + b*x)/(c + d*x)]*(-A + B*n - B*(Log[e*((a + b*x)/(c + d*x))^n] - n*L og[(a + b*x)/(c + d*x)])))/((b*c - a*d)*g*i^2*(c + d*x)) + (Log[(a + b*x)/ (c + d*x)]^2*(A*b*B*c*n - a*B^2*d*n^2 + A*b*B*d*n*x - b*B^2*d*n^2*x + b*B^ 2*c*n*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + b*B^ 2*d*n*x*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])))/(( b*c - a*d)^2*g*i^2*(c + d*x)) + (A^2 - 2*A*B*n + 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)]) - 2*B^2*n*(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)/((b*c - a*d)*g*i^2*(c + d*x)) + (b*Log[a + b*x]*(A^2 - 2*A*B*n + 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)]) - 2*B^2*n*(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))/((b*c - a*d)^2*g*i^2) - (b*(A^2 - 2*A*B*n + 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)]) - 2*B^2*n*(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)^2*g*i^2)
Time = 0.53 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2961, 2788, 2733, 2009, 2739, 15}
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.
| \(\displaystyle \int \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2d\frac {a+b x}{c+d x}}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )d\frac {a+b x}{c+d x}\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {\frac {b \int \frac {(a+b x)^2}{(c+d x)^2}d\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {b (a+b x)^3}{3 B n (c+d x)^3}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
((b*(a + b*x)^3)/(3*B*n*(c + d*x)^3) - d*(((a + b*x)*(A + B*Log[e*((a + b* x)/(c + d*x))^n])^2)/(c + d*x) - 2*B*n*((A*(a + b*x))/(c + d*x) - (B*n*(a + b*x))/(c + d*x) + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(c + d*x) )))/((b*c - a*d)^2*g*i^2)
3.2.98.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
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. \(508\) vs. \(2(229)=458\).
Time = 5.06 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.20
| method | result | size |
| parallelrisch | \(\frac {-6 B^{2} a \,b^{2} d^{4} n^{3}+6 B^{2} b^{3} c \,d^{3} n^{3}-3 A^{2} a \,b^{2} d^{4} n +3 A^{2} b^{3} c \,d^{3} n +B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3} b^{3} d^{4}+B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3} b^{3} c \,d^{3}+3 A^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} d^{4}+3 A^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} c \,d^{3}-6 A B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} d^{4} n -6 A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{2} d^{4} n +6 A B a \,b^{2} d^{4} n^{2}-6 A B \,b^{3} c \,d^{3} n^{2}-3 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{3} d^{4} n +6 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} d^{4} n^{2}+3 A B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{3} d^{4}-3 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{2} d^{4} n +6 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{2} d^{4} n^{2}+3 A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{3} c \,d^{3}}{3 i^{2} g \left (d x +c \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} d^{3} n}\) | \(509\) |
1/3*(-6*B^2*a*b^2*d^4*n^3+6*B^2*b^3*c*d^3*n^3-3*A^2*a*b^2*d^4*n+3*A^2*b^3* c*d^3*n+B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*b^3*d^4+B^2*ln(e*((b*x+a)/(d*x+c ))^n)^3*b^3*c*d^3+3*A^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^3*d^4+3*A^2*ln(e*((b *x+a)/(d*x+c))^n)*b^3*c*d^3-6*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^3*d^4*n-6* A*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^2*d^4*n+6*A*B*a*b^2*d^4*n^2-6*A*B*b^3*c* d^3*n^2-3*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^3*d^4*n+6*B^2*x*ln(e*((b*x+a )/(d*x+c))^n)*b^3*d^4*n^2+3*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^3*d^4-3*B^ 2*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^2*d^4*n+6*B^2*ln(e*((b*x+a)/(d*x+c))^n)* a*b^2*d^4*n^2+3*A*B*ln(e*((b*x+a)/(d*x+c))^n)^2*b^3*c*d^3)/i^2/g/(d*x+c)/( a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/d^3/n
Time = 0.36 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.87 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {3 \, A^{2} b c - 3 \, A^{2} a d + {\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 6 \, {\left (B^{2} b c - B^{2} a d\right )} n^{2} + 3 \, {\left (B^{2} b c - B^{2} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right )^{2} - 3 \, {\left (B^{2} a d n^{2} - A B b c n + {\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (A B b c - A B a d\right )} n + 3 \, {\left (2 \, A B b c - 2 \, A B a d + {\left (B^{2} b d n x + B^{2} b c n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (B^{2} b c - B^{2} a d\right )} n - 2 \, {\left (B^{2} a d n - A B b c + {\left (B^{2} b d n - A B b d\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 3 \, {\left (2 \, B^{2} a d n^{2} - 2 \, A B a d n + A^{2} b c + {\left (2 \, B^{2} b d n^{2} - 2 \, A B b d n + A^{2} b d\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]
1/3*(3*A^2*b*c - 3*A^2*a*d + (B^2*b*d*n^2*x + B^2*b*c*n^2)*log((b*x + a)/( d*x + c))^3 + 6*(B^2*b*c - B^2*a*d)*n^2 + 3*(B^2*b*c - B^2*a*d + (B^2*b*d* x + B^2*b*c)*log((b*x + a)/(d*x + c)))*log(e)^2 - 3*(B^2*a*d*n^2 - A*B*b*c *n + (B^2*b*d*n^2 - A*B*b*d*n)*x)*log((b*x + a)/(d*x + c))^2 - 6*(A*B*b*c - A*B*a*d)*n + 3*(2*A*B*b*c - 2*A*B*a*d + (B^2*b*d*n*x + B^2*b*c*n)*log((b *x + a)/(d*x + c))^2 - 2*(B^2*b*c - B^2*a*d)*n - 2*(B^2*a*d*n - A*B*b*c + (B^2*b*d*n - A*B*b*d)*x)*log((b*x + a)/(d*x + c)))*log(e) + 3*(2*B^2*a*d*n ^2 - 2*A*B*a*d*n + A^2*b*c + (2*B^2*b*d*n^2 - 2*A*B*b*d*n + A^2*b*d)*x)*lo g((b*x + a)/(d*x + c)))/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*g*i^2*x + (b^ 2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*g*i^2)
Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (229) = 458\).
Time = 0.26 (sec) , antiderivative size = 1014, normalized size of antiderivative = 4.39 \[ \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) (c i+d i x)^2} \, dx=B^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2} + 2 \, A B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, {\left (\frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{3} - {\left (b d x + b c\right )} \log \left (d x + c\right )^{3} + 3 \, {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, b c - 6 \, a d + 6 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d x + {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 2 \, b c + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} - \frac {3 \, {\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x}\right )} B^{2} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B n}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} + A^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \]
B^2*(1/((b*c*d - a*d^2)*g*i^2*x + (b*c^2 - a*c*d)*g*i^2) + b*log(b*x + a)/ ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g*i^2) - b*log(d*x + c)/((b^2*c^2 - 2*a*b *c*d + a^2*d^2)*g*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2 + 2*A*B*( 1/((b*c*d - a*d^2)*g*i^2*x + (b*c^2 - a*c*d)*g*i^2) + b*log(b*x + a)/((b^2 *c^2 - 2*a*b*c*d + a^2*d^2)*g*i^2) - b*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*(((b*d*x + b*c)*log(b*x + a)^3 - (b*d*x + b*c)*log(d*x + c)^3 + 3*(b*d*x + b*c)*log( b*x + a)^2 + 3*(b*d*x + b*c + (b*d*x + b*c)*log(b*x + a))*log(d*x + c)^2 + 6*b*c - 6*a*d + 6*(b*d*x + b*c)*log(b*x + a) - 3*(2*b*d*x + (b*d*x + b*c) *log(b*x + a)^2 + 2*b*c + 2*(b*d*x + b*c)*log(b*x + a))*log(d*x + c))*n^2/ (b^2*c^3*g*i^2 - 2*a*b*c^2*d*g*i^2 + a^2*c*d^2*g*i^2 + (b^2*c^2*d*g*i^2 - 2*a*b*c*d^2*g*i^2 + a^2*d^3*g*i^2)*x) - 3*((b*d*x + b*c)*log(b*x + a)^2 + (b*d*x + b*c)*log(d*x + c)^2 + 2*b*c - 2*a*d + 2*(b*d*x + b*c)*log(b*x + a ) - 2*(b*d*x + b*c + (b*d*x + b*c)*log(b*x + a))*log(d*x + c))*n*log(e*(b* x/(d*x + c) + a/(d*x + c))^n)/(b^2*c^3*g*i^2 - 2*a*b*c^2*d*g*i^2 + a^2*c*d ^2*g*i^2 + (b^2*c^2*d*g*i^2 - 2*a*b*c*d^2*g*i^2 + a^2*d^3*g*i^2)*x))*B^2 - ((b*d*x + b*c)*log(b*x + a)^2 + (b*d*x + b*c)*log(d*x + c)^2 + 2*b*c - 2* a*d + 2*(b*d*x + b*c)*log(b*x + a) - 2*(b*d*x + b*c + (b*d*x + b*c)*log(b* x + a))*log(d*x + c))*A*B*n/(b^2*c^3*g*i^2 - 2*a*b*c^2*d*g*i^2 + a^2*c*d^2 *g*i^2 + (b^2*c^2*d*g*i^2 - 2*a*b*c*d^2*g*i^2 + a^2*d^3*g*i^2)*x) + A^2...
Time = 0.90 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.59 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {1}{3} \, {\left (\frac {B^{2} b n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3}}{b c g i^{2} - a d g i^{2}} - 3 \, {\left (\frac {{\left (b x + a\right )} B^{2} d n^{2}}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}} - \frac {B^{2} b n \log \left (e\right ) + A B b n}{b c g i^{2} - a d g i^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + \frac {3 \, {\left (B^{2} b \log \left (e\right )^{2} + 2 \, A B b \log \left (e\right ) + A^{2} b\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b c g i^{2} - a d g i^{2}} + \frac {6 \, {\left (B^{2} d n^{2} - B^{2} d n \log \left (e\right ) - A B d n\right )} {\left (b x + a\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (2 \, B^{2} d n^{2} - 2 \, B^{2} d n \log \left (e\right ) + B^{2} d \log \left (e\right )^{2} - 2 \, A B d n + 2 \, A B d \log \left (e\right ) + A^{2} d\right )} {\left (b x + a\right )}}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
1/3*(B^2*b*n^2*log((b*x + a)/(d*x + c))^3/(b*c*g*i^2 - a*d*g*i^2) - 3*((b* x + a)*B^2*d*n^2/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)) - (B^2*b*n*log(e) + A *B*b*n)/(b*c*g*i^2 - a*d*g*i^2))*log((b*x + a)/(d*x + c))^2 + 3*(B^2*b*log (e)^2 + 2*A*B*b*log(e) + A^2*b)*log((b*x + a)/(d*x + c))/(b*c*g*i^2 - a*d* g*i^2) + 6*(B^2*d*n^2 - B^2*d*n*log(e) - A*B*d*n)*(b*x + a)*log((b*x + a)/ (d*x + c))/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)) - 3*(2*B^2*d*n^2 - 2*B^2*d* n*log(e) + B^2*d*log(e)^2 - 2*A*B*d*n + 2*A*B*d*log(e) + A^2*d)*(b*x + a)/ ((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^ 2)
Time = 1.96 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {B^2\,b\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g\,i^2\,n\,{\left (a\,d-b\,c\right )}^2}-\frac {A^2-2\,A\,B\,n+2\,B^2\,n^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {2\,B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (A-B\,n\right )}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {B\,b\,\left (A-B\,n\right )}{g\,i^2\,n\,{\left (a\,d-b\,c\right )}^2}\right )-\frac {b\,\mathrm {atan}\left (\frac {b\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,\left (A^2-2\,A\,B\,n+2\,B^2\,n^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (b\,A^2-2\,b\,A\,B\,n+2\,b\,B^2\,n^2\right )}\right )\,\left (A^2-2\,A\,B\,n+2\,B^2\,n^2\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]
(B^2*b*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g*i^2*n*(a*d - b*c)^2) - (A^2 + 2*B^2*n^2 - 2*A*B*n)/((a*d - b*c)*(c*g*i^2 + d*g*i^2*x)) - (2*B*log(e*(( a + b*x)/(c + d*x))^n)*(A - B*n))/((a*d - b*c)*(c*g*i^2 + d*g*i^2*x)) - (b *atan((b*(2*b*d*x + (a^2*d^2*g*i^2 - b^2*c^2*g*i^2)/(g*i^2*(a*d - b*c)))*( A^2 + 2*B^2*n^2 - 2*A*B*n)*1i)/((a*d - b*c)*(A^2*b + 2*B^2*b*n^2 - 2*A*B*b *n)))*(A^2 + 2*B^2*n^2 - 2*A*B*n)*2i)/(g*i^2*(a*d - b*c)^2) - log(e*((a + b*x)/(c + d*x))^n)^2*(B^2/((a*d - b*c)*(c*g*i^2 + d*g*i^2*x)) - (B*b*(A - B*n))/(g*i^2*n*(a*d - b*c)^2))