Integrand size = 40, antiderivative size = 365 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {B d^3 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}+\frac {3 b^2 B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^2 i^3}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3} \]
1/4*B*d^3*(b*x+a)^2/(-a*d+b*c)^4/g^2/i^3/(d*x+c)^2-3*b*B*d^2*(b*x+a)/(-a*d +b*c)^4/g^2/i^3/(d*x+c)-b^3*B*(d*x+c)/(-a*d+b*c)^4/g^2/i^3/(b*x+a)+3/2*b^2 *B*d*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^4/g^2/i^3-1/2*d^3*(b*x+a)^2*(A+B*ln( e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^2/i^3/(d*x+c)^2+3*b*d^2*(b*x+a)*(A+B*ln (e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^2/i^3/(d*x+c)-b^3*(d*x+c)*(A+B*ln(e*(b *x+a)/(d*x+c)))/(-a*d+b*c)^4/g^2/i^3/(b*x+a)-3*b^2*d*ln((b*x+a)/(d*x+c))*( A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^2/i^3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.24 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {-\frac {4 b^3 B c}{a+b x}+\frac {4 a b^2 B d}{a+b x}+\frac {B d (b c-a d)^2}{(c+d x)^2}+\frac {8 b^2 B c d}{c+d x}-\frac {8 a b B d^2}{c+d x}+\frac {2 b B d (b c-a d)}{c+d x}+6 b^2 B d \log (a+b x)-\frac {4 b^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {2 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}-\frac {8 b d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-12 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 b^2 B d \log (c+d x)+12 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+6 b^2 B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-6 b^2 B d \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^2 i^3} \]
((-4*b^3*B*c)/(a + b*x) + (4*a*b^2*B*d)/(a + b*x) + (B*d*(b*c - a*d)^2)/(c + d*x)^2 + (8*b^2*B*c*d)/(c + d*x) - (8*a*b*B*d^2)/(c + d*x) + (2*b*B*d*( b*c - a*d))/(c + d*x) + 6*b^2*B*d*Log[a + b*x] - (4*b^2*(b*c - a*d)*(A + B *Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (2*d*(b*c - a*d)^2*(A + B*Log[ (e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 - (8*b*d*(b*c - a*d)*(A + B*Log[(e* (a + b*x))/(c + d*x)]))/(c + d*x) - 12*b^2*d*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*b^2*B*d*Log[c + d*x] + 12*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 6*b^2*B*d*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 6*b^2*B*d*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*L og[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(4*(b*c - a*d)^4*g ^2*i^3)
Time = 0.43 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 2772, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 i^3 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B \int \left (-\frac {(c+d x)^2 b^3}{(a+b x)^2}-\frac {3 d (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) b^2}{a+b x}+3 d^2 b-\frac {d^3 (a+b x)}{2 (c+d x)}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}}{g^2 i^3 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-B \left (\frac {b^3 (c+d x)}{a+b x}-\frac {3}{2} b^2 d \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {d^3 (a+b x)^2}{4 (c+d x)^2}+\frac {3 b d^2 (a+b x)}{c+d x}\right )}{g^2 i^3 (b c-a d)^4}\) |
(-(B*(-1/4*(d^3*(a + b*x)^2)/(c + d*x)^2 + (3*b*d^2*(a + b*x))/(c + d*x) + (b^3*(c + d*x))/(a + b*x) - (3*b^2*d*Log[(a + b*x)/(c + d*x)]^2)/2)) - (d ^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(c + d*x)^2) + (3* b*d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - (b^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - 3*b^2*d*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^2*i ^3)
3.1.52.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
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 = 2.83 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.39
| method | result | size |
| parts | \(\frac {A \left (-\frac {d}{2 \left (a d -c b \right )^{2} \left (d x +c \right )^{2}}+\frac {3 d \,b^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{4}}+\frac {2 d b}{\left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {b^{2}}{\left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {3 d \,b^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{4}}\right )}{g^{2} i^{3}}-\frac {B d \left (\frac {d^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2}}-\frac {3 d b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{2}}+\frac {3 b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2}}-\frac {b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{d \left (a d -c b \right )^{2}}\right )}{g^{2} i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(506\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3}}{i^{3} \left (a d -c b \right )^{5} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d^{3} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {d^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {3 d^{3} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}\right )}{d^{2}}\) | \(632\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3}}{i^{3} \left (a d -c b \right )^{5} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d^{3} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {d^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {3 d^{3} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}\right )}{d^{2}}\) | \(632\) |
| risch | \(-\frac {A d}{2 g^{2} i^{3} \left (a d -c b \right )^{2} \left (d x +c \right )^{2}}+\frac {3 A d \,b^{2} \ln \left (d x +c \right )}{g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {2 A d b}{g^{2} i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {A \,b^{2}}{g^{2} i^{3} \left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {3 A d \,b^{2} \ln \left (b x +a \right )}{g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {5 B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {2 B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b a}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}-\frac {2 B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2} c}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}+\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c b}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2} b^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {11 B d \,b^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4}}-\frac {5 B \,d^{2} b a}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}+\frac {5 B d \,b^{2} c}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}+\frac {B \,d^{3} a^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {B \,d^{2} a c b}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}+\frac {B d \,c^{2} b^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {3 B d \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4}}-\frac {B e \,b^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B e \,b^{3}}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\) | \(869\) |
| parallelrisch | \(-\frac {6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} d^{7}+12 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c \,d^{6}+12 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{7}+24 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c \,d^{6}-18 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{7}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{2} d^{5}+12 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{5}-6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{4} d^{7}+12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{5}+2 A \,a^{3} b^{3} d^{7}+4 A \,b^{6} c^{3} d^{4}-B \,a^{3} b^{3} d^{7}+4 B \,b^{6} c^{3} d^{4}+6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} d^{7}+12 A \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{7}-6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{7}-12 A \,x^{2} a \,b^{5} d^{7}+12 A \,x^{2} b^{6} c \,d^{6}+6 B \,x^{2} a \,b^{5} d^{7}-6 B \,x^{2} b^{6} c \,d^{6}-6 A x \,a^{2} b^{4} d^{7}+18 A x \,b^{6} c^{2} d^{5}+9 B x \,a^{2} b^{4} d^{7}-3 B x \,b^{6} c^{2} d^{5}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{3} d^{7}+4 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d^{4}-12 a^{2} A \,b^{4} c \,d^{6}+6 a A \,b^{5} c^{2} d^{5}+12 a^{2} B \,b^{4} c \,d^{6}-15 a B \,b^{5} c^{2} d^{5}+6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c^{2} d^{5}-12 A x a \,b^{5} c \,d^{6}+12 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{5}-6 B x a \,b^{5} c \,d^{6}-12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{4} c \,d^{6}+12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c \,d^{6}+24 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{6}-24 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{6}}{4 i^{3} g^{2} \left (d x +c \right )^{2} \left (b x +a \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3} d^{4} \left (a d -c b \right )}\) | \(870\) |
| norman | \(\text {Expression too large to display}\) | \(1178\) |
A/g^2/i^3*(-1/2*d/(a*d-b*c)^2/(d*x+c)^2+3*d/(a*d-b*c)^4*b^2*ln(d*x+c)+2*d/ (a*d-b*c)^3*b/(d*x+c)+b^2/(a*d-b*c)^3/(b*x+a)-3*d/(a*d-b*c)^4*b^2*ln(b*x+a ))-B/g^2/i^3*d/(a*d-b*c)^2/e^2*(d^2/(a*d-b*c)^2*(1/2*(b*e/d+(a*d-b*c)*e/d/ (d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4*(b*e/d+(a*d-b*c)*e/d/(d*x+c ))^2)-3*d/(a*d-b*c)^2*b*e*((b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c )*e/d/(d*x+c))-(a*d-b*c)*e/d/(d*x+c)-b*e/d)+3/2/(a*d-b*c)^2*b^2*e^2*ln(b*e /d+(a*d-b*c)*e/d/(d*x+c))^2-1/d/(a*d-b*c)^2*b^3*e^3*(-1/(b*e/d+(a*d-b*c)*e /d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c) )))
Time = 0.31 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.84 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {4 \, {\left (A + B\right )} b^{3} c^{3} + 3 \, {\left (2 \, A - 5 \, B\right )} a b^{2} c^{2} d - 12 \, {\left (A - B\right )} a^{2} b c d^{2} + {\left (2 \, A - B\right )} a^{3} d^{3} + 6 \, {\left ({\left (2 \, A - B\right )} b^{3} c d^{2} - {\left (2 \, A - B\right )} a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left ({\left (6 \, A - B\right )} b^{3} c^{2} d - 2 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} - {\left (2 \, A - 3 \, B\right )} a^{2} b d^{3}\right )} x + 2 \, {\left (3 \, {\left (2 \, A - B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 3 \, {\left (4 \, A b^{3} c d^{2} + {\left (2 \, A - 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (2 \, {\left (A + B\right )} b^{3} c^{2} d + 4 \, {\left (A - B\right )} a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]
-1/4*(4*(A + B)*b^3*c^3 + 3*(2*A - 5*B)*a*b^2*c^2*d - 12*(A - B)*a^2*b*c*d ^2 + (2*A - B)*a^3*d^3 + 6*((2*A - B)*b^3*c*d^2 - (2*A - B)*a*b^2*d^3)*x^2 + 6*(B*b^3*d^3*x^3 + B*a*b^2*c^2*d + (2*B*b^3*c*d^2 + B*a*b^2*d^3)*x^2 + (B*b^3*c^2*d + 2*B*a*b^2*c*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 + 3*((6* A - B)*b^3*c^2*d - 2*(2*A + B)*a*b^2*c*d^2 - (2*A - 3*B)*a^2*b*d^3)*x + 2* (3*(2*A - B)*b^3*d^3*x^3 + 2*B*b^3*c^3 + 6*A*a*b^2*c^2*d - 6*B*a^2*b*c*d^2 + B*a^3*d^3 + 3*(4*A*b^3*c*d^2 + (2*A - 3*B)*a*b^2*d^3)*x^2 + 3*(2*(A + B )*b^3*c^2*d + 4*(A - B)*a*b^2*c*d^2 - B*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d *x + c)))/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2* c*d^5 + a^4*b*d^6)*g^2*i^3*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^ 3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*g^2*i^3*x^2 + (b^ 5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^ 2*d^4 + 2*a^5*c*d^5)*g^2*i^3*x + (a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2* c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4)*g^2*i^3)
Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1721 vs. \(2 (359) = 718\).
Time = 0.33 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \]
-1/2*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5) *g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b* c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^ 3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/2* A*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d ^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2* i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a ^3*b*c^2*d^3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a ^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4*c^4 - 4* a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^ 2*d*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c *d^3 + a^4*d^4)*g^2*i^3)) - 1/4*(4*b^3*c^3 - 15*a*b^2*c^2*d + 12*a^2*b*c*d ^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2* d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(...
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3}} \,d x } \]
Time = 5.71 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.69 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {A\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {A\,a^2\,d^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}+\frac {B\,a^2\,d^2}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {B\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,b^2\,d^2\,x^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d^2\,x^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {5\,A\,a\,b\,c\,d}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {11\,B\,a\,b\,c\,d}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,a\,b\,d^2\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {9\,B\,a\,b\,d^2\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {9\,A\,b^2\,c\,d\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,c\,d\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,6{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,b^2\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \]
(A*b^2*d*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*6i)/(g^2*i^3*(a*d - b*c)^4) - (3*B*b^2*d*log((e*(a + b*x))/(c + d*x))^2)/(2*g^2*i^3*(a*d - b *c)^4) - (B*b^2*d*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*3i)/(g^2* i^3*(a*d - b*c)^4) - (A*a^2*d^2)/(2*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d *x)^2) + (A*b^2*c^2)/(g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (B*a^ 2*d^2)/(4*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (B*b^2*c^2)/(g^2* i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (B*a*d*log((e*(a + b*x))/(c + d *x)))/(2*g^2*i^3*(a*d - b*c)^2*(a + b*x)*(c + d*x)^2) - (B*b*c*log((e*(a + b*x))/(c + d*x)))/(g^2*i^3*(a*d - b*c)^2*(a + b*x)*(c + d*x)^2) + (3*A*b^ 2*d^2*x^2)/(g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (3*B*b^2*d^2*x^ 2)/(2*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (5*A*a*b*c*d)/(2*g^2* i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (11*B*a*b*c*d)/(4*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (3*B*b*d*x*log((e*(a + b*x))/(c + d*x))) /(2*g^2*i^3*(a*d - b*c)^2*(a + b*x)*(c + d*x)^2) + (3*B*b^2*d^2*x^2*log((e *(a + b*x))/(c + d*x)))/(g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (3 *A*a*b*d^2*x)/(2*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (9*B*a*b*d ^2*x)/(4*g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (9*A*b^2*c*d*x)/(2 *g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) - (3*B*b^2*c*d*x)/(4*g^2*i^3 *(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (3*B*a*b*c*d*log((e*(a + b*x))/(c + d*x)))/(g^2*i^3*(a*d - b*c)^3*(a + b*x)*(c + d*x)^2) + (3*B*a*b*d^2*x...