Integrand size = 40, antiderivative size = 463 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=-\frac {B d^4 (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {4 b B d^3 (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 B d (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}-\frac {3 b^2 B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3} \]
-1/4*B*d^4*(b*x+a)^2/(-a*d+b*c)^5/g^3/i^3/(d*x+c)^2+4*b*B*d^3*(b*x+a)/(-a* d+b*c)^5/g^3/i^3/(d*x+c)+4*b^3*B*d*(d*x+c)/(-a*d+b*c)^5/g^3/i^3/(b*x+a)-1/ 4*b^4*B*(d*x+c)^2/(-a*d+b*c)^5/g^3/i^3/(b*x+a)^2-3*b^2*B*d^2*ln((b*x+a)/(d *x+c))^2/(-a*d+b*c)^5/g^3/i^3+1/2*d^4*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)) )/(-a*d+b*c)^5/g^3/i^3/(d*x+c)^2-4*b*d^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c) ))/(-a*d+b*c)^5/g^3/i^3/(d*x+c)+4*b^3*d*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)) )/(-a*d+b*c)^5/g^3/i^3/(b*x+a)-1/2*b^4*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c) ))/(-a*d+b*c)^5/g^3/i^3/(b*x+a)^2+6*b^2*d^2*ln((b*x+a)/(d*x+c))*(A+B*ln(e* (b*x+a)/(d*x+c)))/(-a*d+b*c)^5/g^3/i^3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.15 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=-\frac {\frac {b^2 B (b c-a d)^2}{(a+b x)^2}-\frac {12 b^3 B c d}{a+b x}+\frac {12 a b^2 B d^2}{a+b x}-\frac {2 b^2 B d (b c-a d)}{a+b x}+\frac {B d^2 (b c-a d)^2}{(c+d x)^2}+\frac {12 b^2 B c d^2}{c+d x}-\frac {12 a b B d^3}{c+d x}+\frac {2 b B d^2 (b c-a d)}{c+d x}+\frac {2 b^2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}-\frac {12 b^2 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {2 d^2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}-\frac {12 b d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-24 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+12 b^2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-12 b^2 B d^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^5 g^3 i^3} \]
-1/4*((b^2*B*(b*c - a*d)^2)/(a + b*x)^2 - (12*b^3*B*c*d)/(a + b*x) + (12*a *b^2*B*d^2)/(a + b*x) - (2*b^2*B*d*(b*c - a*d))/(a + b*x) + (B*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (12*b^2*B*c*d^2)/(c + d*x) - (12*a*b*B*d^3)/(c + d*x ) + (2*b*B*d^2*(b*c - a*d))/(c + d*x) + (2*b^2*(b*c - a*d)^2*(A + B*Log[(e *(a + b*x))/(c + d*x)]))/(a + b*x)^2 - (12*b^2*d*(b*c - a*d)*(A + B*Log[(e *(a + b*x))/(c + d*x)]))/(a + b*x) - (2*d^2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 - (12*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - 24*b^2*d^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*b^2*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log [c + d*x] + 12*b^2*B*d^2*(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)]) - 12*b^2*B*d^ 2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*P olyLog[2, (b*(c + d*x))/(b*c - a*d)]))/((b*c - a*d)^5*g^3*i^3)
Time = 0.48 (sec) , antiderivative size = 319, 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.
\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^3 (c i+d i x)^3} \, dx\) |
\(\Big \downarrow \) 2962 |
\(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i^3 (b c-a d)^5}\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle \frac {-B \int \left (\frac {6 b^2 (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) d^2}{a+b x}+\frac {1}{2} \left (-\frac {(c+d x)^3 b^4}{(a+b x)^3}+\frac {8 d (c+d x)^2 b^3}{(a+b x)^2}-8 d^3 b+\frac {d^4 (a+b x)}{c+d x}\right )\right )d\frac {a+b x}{c+d x}-\frac {b^4 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {4 b^3 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {d^4 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}}{g^3 i^3 (b c-a d)^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {b^4 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {4 b^3 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {d^4 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-B \left (\frac {b^4 (c+d x)^2}{4 (a+b x)^2}-\frac {4 b^3 d (c+d x)}{a+b x}+3 b^2 d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+\frac {d^4 (a+b x)^2}{4 (c+d x)^2}-\frac {4 b d^3 (a+b x)}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}\) |
(-(B*((d^4*(a + b*x)^2)/(4*(c + d*x)^2) - (4*b*d^3*(a + b*x))/(c + d*x) - (4*b^3*d*(c + d*x))/(a + b*x) + (b^4*(c + d*x)^2)/(4*(a + b*x)^2) + 3*b^2* d^2*Log[(a + b*x)/(c + d*x)]^2)) + (d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x ))/(c + d*x)]))/(2*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[(e*(a + b* x))/(c + d*x)]))/(c + d*x) + (4*b^3*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/( c + d*x)]))/(a + b*x) - (b^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x )]))/(2*(a + b*x)^2) + 6*b^2*d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3)
3.1.53.3.1 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 = 2.08 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.40
method | result | size |
parts | \(\frac {A \left (-\frac {d^{2}}{2 \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}+\frac {6 d^{2} b^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{5}}+\frac {3 d^{2} b}{\left (a d -c b \right )^{4} \left (d x +c \right )}+\frac {b^{2}}{2 \left (a d -c b \right )^{3} \left (b x +a \right )^{2}}-\frac {6 d^{2} b^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{5}}+\frac {3 b^{2} d}{\left (a d -c b \right )^{4} \left (b x +a \right )}\right )}{g^{3} i^{3}}-\frac {B d \left (\frac {d^{3} \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 )^{3}}-\frac {4 d^{2} 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 )^{3}}+\frac {3 d \,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}}{\left (a d -c b \right )^{3}}-\frac {4 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 )}{\left (a d -c b \right )^{3}}+\frac {b^{4} e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{d \left (a d -c b \right )^{3}}\right )}{g^{3} i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(649\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e A \,b^{4}}{2 i^{3} \left (a d -c b \right )^{6} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {4 d^{3} A \,b^{3}}{i^{3} \left (a d -c b \right )^{6} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {6 d^{4} 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 )^{6} g^{3}}-\frac {4 d^{5} 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 )^{6} g^{3}}+\frac {d^{6} 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 )^{6} g^{3}}+\frac {d^{2} e B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{3} \left (a d -c b \right )^{6} g^{3}}-\frac {4 d^{3} 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 )^{6} g^{3}}+\frac {3 d^{4} 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}}{e \,i^{3} \left (a d -c b \right )^{6} g^{3}}-\frac {4 d^{5} 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 )^{6} g^{3}}+\frac {d^{6} 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 )^{6} g^{3}}\right )}{d^{2}}\) | \(804\) |
default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e A \,b^{4}}{2 i^{3} \left (a d -c b \right )^{6} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {4 d^{3} A \,b^{3}}{i^{3} \left (a d -c b \right )^{6} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {6 d^{4} 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 )^{6} g^{3}}-\frac {4 d^{5} 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 )^{6} g^{3}}+\frac {d^{6} 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 )^{6} g^{3}}+\frac {d^{2} e B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{3} \left (a d -c b \right )^{6} g^{3}}-\frac {4 d^{3} 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 )^{6} g^{3}}+\frac {3 d^{4} 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}}{e \,i^{3} \left (a d -c b \right )^{6} g^{3}}-\frac {4 d^{5} 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 )^{6} g^{3}}+\frac {d^{6} 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 )^{6} g^{3}}\right )}{d^{2}}\) | \(804\) |
risch | \(\text {Expression too large to display}\) | \(1070\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1292\) |
norman | \(\text {Expression too large to display}\) | \(1722\) |
A/g^3/i^3*(-1/2*d^2/(a*d-b*c)^3/(d*x+c)^2+6*d^2/(a*d-b*c)^5*b^2*ln(d*x+c)+ 3*d^2/(a*d-b*c)^4*b/(d*x+c)+1/2*b^2/(a*d-b*c)^3/(b*x+a)^2-6*d^2/(a*d-b*c)^ 5*b^2*ln(b*x+a)+3*b^2/(a*d-b*c)^4*d/(b*x+a))-B/g^3/i^3*d/(a*d-b*c)^2/e^2*( d^3/(a*d-b*c)^3*(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)-4*d^2/(a*d-b*c)^3*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*d/(a*d-b*c)^3*b^2*e^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-4/ (a*d-b*c)^3*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)))+1/d/(a*d-b*c)^3*b^4*e^4*(-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))
Leaf count of result is larger than twice the leaf count of optimal. 1011 vs. \(2 (455) = 910\).
Time = 0.33 (sec) , antiderivative size = 1011, normalized size of antiderivative = 2.18 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=-\frac {{\left (2 \, A + B\right )} b^{4} c^{4} - 16 \, {\left (A + B\right )} a b^{3} c^{3} d + 30 \, B a^{2} b^{2} c^{2} d^{2} + 16 \, {\left (A - B\right )} a^{3} b c d^{3} - {\left (2 \, A - B\right )} a^{4} d^{4} - 24 \, {\left (A b^{4} c d^{3} - A a b^{3} d^{4}\right )} x^{3} - 12 \, {\left ({\left (3 \, A + B\right )} b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3} - {\left (3 \, A - B\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 12 \, {\left (B b^{4} d^{4} x^{4} + B a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (B b^{4} c d^{3} + B a b^{3} d^{4}\right )} x^{3} + {\left (B b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (B a b^{3} c^{2} d^{2} + B a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 4 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{4} c^{3} d + 3 \, {\left (4 \, A - B\right )} a b^{3} c^{2} d^{2} - 3 \, {\left (4 \, A + B\right )} a^{2} b^{2} c d^{3} - {\left (2 \, A - 3 \, B\right )} a^{3} b d^{4}\right )} x - 2 \, {\left (12 \, A b^{4} d^{4} x^{4} - B b^{4} c^{4} + 8 \, B a b^{3} c^{3} d + 12 \, A a^{2} b^{2} c^{2} d^{2} - 8 \, B a^{3} b c d^{3} + B a^{4} d^{4} + 12 \, {\left ({\left (2 \, A + B\right )} b^{4} c d^{3} + {\left (2 \, A - B\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{4} c^{2} d^{2} + 8 \, A a b^{3} c d^{3} + {\left (2 \, A - 3 \, B\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (B b^{4} c^{3} d + 6 \, {\left (A + B\right )} a b^{3} c^{2} d^{2} + 6 \, {\left (A - B\right )} a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} g^{3} i^{3} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} g^{3} i^{3} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} g^{3} i^{3} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} g^{3} i^{3} x + {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5}\right )} g^{3} i^{3}\right )}} \]
-1/4*((2*A + B)*b^4*c^4 - 16*(A + B)*a*b^3*c^3*d + 30*B*a^2*b^2*c^2*d^2 + 16*(A - B)*a^3*b*c*d^3 - (2*A - B)*a^4*d^4 - 24*(A*b^4*c*d^3 - A*a*b^3*d^4 )*x^3 - 12*((3*A + B)*b^4*c^2*d^2 - 2*B*a*b^3*c*d^3 - (3*A - B)*a^2*b^2*d^ 4)*x^2 - 12*(B*b^4*d^4*x^4 + B*a^2*b^2*c^2*d^2 + 2*(B*b^4*c*d^3 + B*a*b^3* d^4)*x^3 + (B*b^4*c^2*d^2 + 4*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*x^2 + 2*(B*a* b^3*c^2*d^2 + B*a^2*b^2*c*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 4*((2*A + 3*B)*b^4*c^3*d + 3*(4*A - B)*a*b^3*c^2*d^2 - 3*(4*A + B)*a^2*b^2*c*d^3 - (2*A - 3*B)*a^3*b*d^4)*x - 2*(12*A*b^4*d^4*x^4 - B*b^4*c^4 + 8*B*a*b^3*c ^3*d + 12*A*a^2*b^2*c^2*d^2 - 8*B*a^3*b*c*d^3 + B*a^4*d^4 + 12*((2*A + B)* b^4*c*d^3 + (2*A - B)*a*b^3*d^4)*x^3 + 6*((2*A + 3*B)*b^4*c^2*d^2 + 8*A*a* b^3*c*d^3 + (2*A - 3*B)*a^2*b^2*d^4)*x^2 + 4*(B*b^4*c^3*d + 6*(A + B)*a*b^ 3*c^2*d^2 + 6*(A - B)*a^2*b^2*c*d^3 - B*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d *x + c)))/((b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^ 4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*g^3*i^3*x^4 + 2*(b^7*c^6*d - 4* a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*g^3*i^3*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a ^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^ 7*d^7)*g^3*i^3*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*g^3*i^3*x + (a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c...
Leaf count of result is larger than twice the leaf count of optimal. 2106 vs. \(2 (430) = 860\).
Time = 138.12 (sec) , antiderivative size = 2106, normalized size of antiderivative = 4.55 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
6*A*b**2*d**2*log(x + (-6*A*a**6*b**2*d**8/(a*d - b*c)**5 + 36*A*a**5*b**3 *c*d**7/(a*d - b*c)**5 - 90*A*a**4*b**4*c**2*d**6/(a*d - b*c)**5 + 120*A*a **3*b**5*c**3*d**5/(a*d - b*c)**5 - 90*A*a**2*b**6*c**4*d**4/(a*d - b*c)** 5 + 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d**3 - 6*A*b**8*c**6 *d**2/(a*d - b*c)**5 + 6*A*b**3*c*d**2)/(12*A*b**3*d**3))/(g**3*i**3*(a*d - b*c)**5) - 6*A*b**2*d**2*log(x + (6*A*a**6*b**2*d**8/(a*d - b*c)**5 - 36 *A*a**5*b**3*c*d**7/(a*d - b*c)**5 + 90*A*a**4*b**4*c**2*d**6/(a*d - b*c)* *5 - 120*A*a**3*b**5*c**3*d**5/(a*d - b*c)**5 + 90*A*a**2*b**6*c**4*d**4/( a*d - b*c)**5 - 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d**3 + 6 *A*b**8*c**6*d**2/(a*d - b*c)**5 + 6*A*b**3*c*d**2)/(12*A*b**3*d**3))/(g** 3*i**3*(a*d - b*c)**5) - 3*B*b**2*d**2*log(e*(a + b*x)/(c + d*x))**2/(a**5 *d**5*g**3*i**3 - 5*a**4*b*c*d**4*g**3*i**3 + 10*a**3*b**2*c**2*d**3*g**3* i**3 - 10*a**2*b**3*c**3*d**2*g**3*i**3 + 5*a*b**4*c**4*d*g**3*i**3 - b**5 *c**5*g**3*i**3) + (-B*a**3*d**3 + 7*B*a**2*b*c*d**2 + 4*B*a**2*b*d**3*x + 7*B*a*b**2*c**2*d + 28*B*a*b**2*c*d**2*x + 18*B*a*b**2*d**3*x**2 - B*b**3 *c**3 + 4*B*b**3*c**2*d*x + 18*B*b**3*c*d**2*x**2 + 12*B*b**3*d**3*x**3)*l og(e*(a + b*x)/(c + d*x))/(2*a**6*c**2*d**4*g**3*i**3 + 4*a**6*c*d**5*g**3 *i**3*x + 2*a**6*d**6*g**3*i**3*x**2 - 8*a**5*b*c**3*d**3*g**3*i**3 - 12*a **5*b*c**2*d**4*g**3*i**3*x + 4*a**5*b*d**6*g**3*i**3*x**3 + 12*a**4*b**2* c**4*d**2*g**3*i**3 + 8*a**4*b**2*c**3*d**3*g**3*i**3*x - 18*a**4*b**2*...
Leaf count of result is larger than twice the leaf count of optimal. 2380 vs. \(2 (455) = 910\).
Time = 0.35 (sec) , antiderivative size = 2380, normalized size of antiderivative = 5.14 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
1/2*B*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d ^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d ^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4 *c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*g^3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^ 6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*g^3*i^3*x + (a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)*g^3 *i^3) + 12*b^2*d^2*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3 *d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^3*i^3) - 12*b^2*d^2 *log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2* c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^3*i^3))*log(b*e*x/(d*x + c) + a*e/(d* x + c)) + 1/2*A*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a ^3*b^3*c*d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*g^3* i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^ 2*d^4 + a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b...
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{3} {\left (d i x + c i\right )}^{3}} \,d x } \]
Time = 9.10 (sec) , antiderivative size = 1443, normalized size of antiderivative = 3.12 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
(A*b^2*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*12i)/(g^3*i^3*(a *d - b*c)^5) - (3*B*b^2*d^2*log((e*(a + b*x))/(c + d*x))^2)/(g^3*i^3*(a*d - b*c)^5) - (A*a^3*d^3)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (A*b^3*c^3)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (B*a^3*d ^3)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (B*b^3*c^3)/(4*g^3 *i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (B*a*d*log((e*(a + b*x))/(c + d*x)))/(2*g^3*i^3*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)^2) - (B*b*c*log((e *(a + b*x))/(c + d*x)))/(2*g^3*i^3*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)^2) + (6*A*b^3*d^3*x^3)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (7*A *a*b^2*c^2*d)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (7*A*a^2 *b*c*d^2)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (15*B*a*b^2* c^2*d)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (15*B*a^2*b*c*d ^2)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (2*A*a^2*b*d^3*x)/ (g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (3*B*a^2*b*d^3*x)/(g^3*i ^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (2*A*b^3*c^2*d*x)/(g^3*i^3*(a* d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*b^3*c^2*d*x)/(g^3*i^3*(a*d - b* c)^4*(a + b*x)^2*(c + d*x)^2) + (9*A*a*b^2*d^3*x^2)/(g^3*i^3*(a*d - b*c)^4 *(a + b*x)^2*(c + d*x)^2) - (3*B*a*b^2*d^3*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (9*A*b^3*c*d^2*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b* x)^2*(c + d*x)^2) + (3*B*b^3*c*d^2*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b*x...