Integrand size = 42, antiderivative size = 214 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {2 A B d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}-\frac {2 B^2 d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}+\frac {2 B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^2 g i^2 (c+d x)}-\frac {d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g i^2 (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g i^2} \]
2*A*B*d*(b*x+a)/(-a*d+b*c)^2/g/i^2/(d*x+c)-2*B^2*d*(b*x+a)/(-a*d+b*c)^2/g/ i^2/(d*x+c)+2*B^2*d*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/(-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)))^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)))^3/B/(-a*d+b*c)^2/g/i^2
Time = 0.35 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {3 b \left (A^2-2 A B+2 B^2\right ) (c+d x) \log (a+b x)+6 (A-B) B (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (-B d (a+b x)+A b (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+b B^2 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-3 \left (A^2-2 A B+2 B^2\right ) (-b c+a d+b (c+d x) \log (c+d x))}{3 (b c-a d)^2 g i^2 (c+d x)} \]
(3*b*(A^2 - 2*A*B + 2*B^2)*(c + d*x)*Log[a + b*x] + 6*(A - B)*B*(b*c - a*d )*Log[(e*(a + b*x))/(c + d*x)] + 3*B*(-(B*d*(a + b*x)) + A*b*(c + d*x))*Lo g[(e*(a + b*x))/(c + d*x)]^2 + b*B^2*(c + d*x)*Log[(e*(a + b*x))/(c + d*x) ]^3 - 3*(A^2 - 2*A*B + 2*B^2)*(-(b*c) + a*d + b*(c + d*x)*Log[c + d*x]))/( 3*(b*c - a*d)^2*g*i^2*(c + d*x))
Time = 0.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2962, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2d\frac {a+b x}{c+d x}}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )d\frac {a+b x}{c+d x}\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {B (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {\frac {b \int \frac {(a+b x)^2}{(c+d x)^2}d\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}-d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {B (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {b (a+b x)^3}{3 B (c+d x)^3}-d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {B (a+b x)}{c+d x}\right )\right )}{g i^2 (b c-a d)^2}\) |
((b*(a + b*x)^3)/(3*B*(c + d*x)^3) - d*(((a + b*x)*(A + B*Log[(e*(a + b*x) )/(c + d*x)])^2)/(c + d*x) - 2*B*((A*(a + b*x))/(c + d*x) - (B*(a + b*x))/ (c + d*x) + (B*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(c + d*x))))/((b*c - a*d)^2*g*i^2)
3.1.96.3.1 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_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Time = 0.98 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.86
| method | result | size |
| norman | \(\frac {\frac {\left (A^{2} b c -2 A B a d +2 B^{2} a d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B \left (A b c -B a d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \left (A^{2} b -2 A B b +2 B^{2} b \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (A^{2}-2 B A +2 B^{2}\right ) d x}{g i c \left (a d -c b \right )}+\frac {b d B \left (-B +A \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} b c \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{i \left (d x +c \right )}\) | \(398\) |
| parallelrisch | \(\frac {-3 A^{2} a \,b^{2} d^{4}+3 A^{2} b^{3} c \,d^{3}+6 A B a \,b^{2} d^{4}-6 A B \,b^{3} c \,d^{3}-6 B^{2} a \,b^{2} d^{4}+6 B^{2} b^{3} c \,d^{3}+B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{3} d^{4}-3 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4}+B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{3} c \,d^{3}+3 A^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}+6 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}-3 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{2} d^{4}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{3}+6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4}+3 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4}-6 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} c \,d^{3}-6 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4}}{3 i^{2} g \left (d x +c \right ) b^{2} d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(446\) |
| parts | \(\frac {A^{2} \left (-\frac {1}{\left (a d -c b \right ) \left (d x +c \right )}-\frac {b \ln \left (d x +c \right )}{\left (a d -c b \right )^{2}}+\frac {b \ln \left (b x +a \right )}{\left (a d -c b \right )^{2}}\right )}{g \,i^{2}}-\frac {B^{2} \left (\frac {d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{a d -c b}-\frac {b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (a d -c b \right )}\right )}{g \,i^{2} \left (a d -c b \right ) e}-\frac {2 B A \left (\frac {d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{a d -c b}-\frac {b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )}\right )}{g \,i^{2} \left (a d -c b \right ) e}\) | \(459\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} A B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {2 d^{3} A B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}\right )}{d^{2}}\) | \(521\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} A B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {2 d^{3} A B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}\right )}{d^{2}}\) | \(521\) |
| risch | \(-\frac {A^{2}}{g \,i^{2} \left (a d -c b \right ) \left (d x +c \right )}-\frac {A^{2} b \ln \left (d x +c \right )}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {A^{2} b \ln \left (b x +a \right )}{g \,i^{2} \left (a d -c b \right )^{2}}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} b}{g \,i^{2} \left (a d -c b \right )^{2}}-\frac {B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {2 B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} d a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {2 B^{2} c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} b}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g \,i^{2} \left (a d -c b \right )^{2}}-\frac {2 B A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b}{g \,i^{2} \left (a d -c b \right )^{2}}-\frac {2 B A d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {2 B A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {2 B A d a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {2 B A c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {2 B A b}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {B A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} b}{g \,i^{2} \left (a d -c b \right )^{2}}\) | \(848\) |
((A^2*b*c-2*A*B*a*d+2*B^2*a*d)/g/i/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a )/(d*x+c))+B*(A*b*c-B*a*d)/g/i/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/(d *x+c))^2+1/g/i*d*(A^2*b-2*A*B*b+2*B^2*b)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln( e*(b*x+a)/(d*x+c))+(A^2-2*A*B+2*B^2)*d/g/i/c/(a*d-b*c)*x+b*d*B*(-B+A)/g/i/ (a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^2+1/3*B^2*b*c/g/i/(a^2 *d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/(d*x+c))^3+1/3*b*B^2*d/g/i/(a^2*d^2-2 *a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^3)/i/(d*x+c)
Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (A B b c - B^{2} a d + {\left (A B - B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} b c + {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b d x - 2 \, {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]
1/3*((B^2*b*d*x + B^2*b*c)*log((b*e*x + a*e)/(d*x + c))^3 + 3*(A^2 - 2*A*B + 2*B^2)*b*c - 3*(A^2 - 2*A*B + 2*B^2)*a*d + 3*(A*B*b*c - B^2*a*d + (A*B - B^2)*b*d*x)*log((b*e*x + a*e)/(d*x + c))^2 + 3*(A^2*b*c + (A^2 - 2*A*B + 2*B^2)*b*d*x - 2*(A*B - B^2)*a*d)*log((b*e*x + a*e)/(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)
Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (185) = 370\).
Time = 0.73 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.52 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {B^{2} b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g i^{2} - 6 a b c d g i^{2} + 3 b^{2} c^{2} g i^{2}} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A^{2} - 2 A B + 2 B^{2}\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) + \frac {\left (A B b c + A B b d x - B^{2} a d - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{2} c d^{2} g i^{2} + a^{2} d^{3} g i^{2} x - 2 a b c^{2} d g i^{2} - 2 a b c d^{2} g i^{2} x + b^{2} c^{3} g i^{2} + b^{2} c^{2} d g i^{2} x} \]
B**2*b*log(e*(a + b*x)/(c + d*x))**3/(3*a**2*d**2*g*i**2 - 6*a*b*c*d*g*i** 2 + 3*b**2*c**2*g*i**2) + (-2*A*B + 2*B**2)*log(e*(a + b*x)/(c + d*x))/(a* c*d*g*i**2 + a*d**2*g*i**2*x - b*c**2*g*i**2 - b*c*d*g*i**2*x) + (A**2 - 2 *A*B + 2*B**2)*(-b*log(x + (-a**3*b*d**3/(a*d - b*c)**2 + 3*a**2*b**2*c*d* *2/(a*d - b*c)**2 - 3*a*b**3*c**2*d/(a*d - b*c)**2 + a*b*d + b**4*c**3/(a* d - b*c)**2 + b**2*c)/(2*b**2*d))/(g*i**2*(a*d - b*c)**2) + b*log(x + (a** 3*b*d**3/(a*d - b*c)**2 - 3*a**2*b**2*c*d**2/(a*d - b*c)**2 + 3*a*b**3*c** 2*d/(a*d - b*c)**2 + a*b*d - b**4*c**3/(a*d - b*c)**2 + b**2*c)/(2*b**2*d) )/(g*i**2*(a*d - b*c)**2) - 1/(a*c*d*g*i**2 - b*c**2*g*i**2 + x*(a*d**2*g* i**2 - b*c*d*g*i**2))) + (A*B*b*c + A*B*b*d*x - B**2*a*d - B**2*b*d*x)*log (e*(a + b*x)/(c + d*x))**2/(a**2*c*d**2*g*i**2 + a**2*d**3*g*i**2*x - 2*a* b*c**2*d*g*i**2 - 2*a*b*c*d**2*g*i**2*x + b**2*c**3*g*i**2 + b**2*c**2*d*g *i**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (212) = 424\).
Time = 0.26 (sec) , antiderivative size = 1004, normalized size of antiderivative = 4.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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 (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\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 (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\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 )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\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} - \frac {{\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 )}{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 )} + 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 )} - \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}{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} \]
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(b*e*x/(d*x + c) + a*e/(d*x + c))^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(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/3*B^2*(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))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(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*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))/(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*(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)) - ((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*...
Time = 0.43 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {1}{3} \, {\left (\frac {B^{2} b e \log \left (\frac {b e x + a e}{d x + c}\right )^{3}}{b c g i^{2} - a d g i^{2}} + \frac {3 \, A^{2} b e \log \left (\frac {b e x + a e}{d x + c}\right )}{b c g i^{2} - a d g i^{2}} + 3 \, {\left (\frac {A B b e}{b c g i^{2} - a d g i^{2}} - \frac {{\left (b e x + a e\right )} B^{2} d}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - \frac {6 \, {\left (A B d - B^{2} d\right )} {\left (b e x + a e\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (A^{2} d - 2 \, A B d + 2 \, B^{2} d\right )} {\left (b e x + a e\right )}}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
1/3*(B^2*b*e*log((b*e*x + a*e)/(d*x + c))^3/(b*c*g*i^2 - a*d*g*i^2) + 3*A^ 2*b*e*log((b*e*x + a*e)/(d*x + c))/(b*c*g*i^2 - a*d*g*i^2) + 3*(A*B*b*e/(b *c*g*i^2 - a*d*g*i^2) - (b*e*x + a*e)*B^2*d/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)))*log((b*e*x + a*e)/(d*x + c))^2 - 6*(A*B*d - B^2*d)*(b*e*x + a*e)*lo g((b*e*x + a*e)/(d*x + c))/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)) - 3*(A^2*d - 2*A*B*d + 2*B^2*d)*(b*e*x + a*e)/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)))*(b *c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 2.92 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx={\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,b\,\left (A-B\right )}{g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}+\frac {B^2\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (A-B\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {b\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (b\,A^2-2\,b\,A\,B+2\,b\,B^2\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]
log((e*(a + b*x))/(c + d*x))^2*((B*b*(A - B))/(g*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*(a*d - b*c))/(b*d*g*i^2*(x/b + c/(b*d))*(a^2*d^2 + b^2* c^2 - 2*a*b*c*d))) - (A^2 + 2*B^2 - 2*A*B)/((a*d - b*c)*(c*g*i^2 + d*g*i^2 *x)) + (B^2*b*log((e*(a + b*x))/(c + d*x))^3)/(3*g*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (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 - 2*A*B)*1i)/((a*d - b*c)*(A^2*b + 2*B^2*b - 2*A*B*b)))*(A^2 + 2*B^2 - 2*A*B)*2i)/(g*i^2*(a*d - b*c)^2) - (2*B*log((e* (a + b*x))/(c + d*x))*(A - B)*(a*d - b*c))/(b*d*g*i^2*(x/b + c/(b*d))*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d))