Integrand size = 40, antiderivative size = 261 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {B d^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 B (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {b B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}+\frac {d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^2 i^2} \] Output:
-B*d^2*(b*x+a)/(-a*d+b*c)^3/g^2/i^2/(d*x+c)-b^2*B*(d*x+c)/(-a*d+b*c)^3/g^2 /i^2/(b*x+a)+b*B*d*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^3/g^2/i^2+d^2*(b*x+a)* (A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^2/i^2/(d*x+c)-b^2*(d*x+c)*(A+B* ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^2/i^2/(b*x+a)-2*b*d*ln((b*x+a)/(d*x+ c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^2/i^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.42 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.24 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {-\frac {b^2 B c}{a+b x}+\frac {a b B d}{a+b x}+\frac {b B c d}{c+d x}-\frac {a B d^2}{c+d x}-\frac {b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+\frac {d (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-2 b d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+b B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-b B d \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3 g^2 i^2} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^2*(c*i + d*i *x)^2),x]
Output:
(-((b^2*B*c)/(a + b*x)) + (a*b*B*d)/(a + b*x) + (b*B*c*d)/(c + d*x) - (a*B *d^2)/(c + d*x) - (b*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + (d*(-(b*c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - 2*b*d*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*b*d*(A + B* Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + b*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)]) - b*B*d*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x]) *Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/((b*c - a*d)^3*g ^2*i^2)
Time = 0.42 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.70, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 i^2 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B \int \left (d^2-\frac {2 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) d}{a+b x}-\frac {b^2 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-2 b d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-2 b d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-B \left (\frac {b^2 (c+d x)}{a+b x}+\frac {d^2 (a+b x)}{c+d x}-b d \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{g^2 i^2 (b c-a d)^3}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^2*(c*i + d*i*x)^2) ,x]
Output:
(-(B*((d^2*(a + b*x))/(c + d*x) + (b^2*(c + d*x))/(a + b*x) - b*d*Log[(a + b*x)/(c + d*x)]^2)) + (d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) )/(c + d*x) - (b^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b* x) - 2*b*d*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/ ((b*c - a*d)^3*g^2*i^2)
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 = 1.88 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.40
| method | result | size |
| parts | \(\frac {A \left (-\frac {d}{\left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 d b \ln \left (d x +c \right )}{\left (d a -b c \right )^{3}}-\frac {b}{\left (d a -b c \right )^{2} \left (b x +a \right )}+\frac {2 d b \ln \left (b x +a \right )}{\left (d a -b c \right )^{3}}\right )}{g^{2} i^{2}}-\frac {B \left (\frac {d^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2}}-\frac {b d e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\left (d a -b c \right )^{2}}+\frac {b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{2}}\right )}{g^{2} i^{2} \left (d a -b c \right ) e}\) | \(366\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A \,b^{2}}{i^{2} \left (d a -b c \right )^{4} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{3} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{2} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {d^{3} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}\right )}{d^{2}}\) | \(455\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A \,b^{2}}{i^{2} \left (d a -b c \right )^{4} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{3} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{2} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {d^{3} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}\right )}{d^{2}}\) | \(455\) |
| risch | \(-\frac {A d}{g^{2} i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 A d b \ln \left (d x +c \right )}{g^{2} i^{2} \left (d a -b c \right )^{3}}-\frac {A b}{g^{2} i^{2} \left (d a -b c \right )^{2} \left (b x +a \right )}+\frac {2 A d b \ln \left (b x +a \right )}{g^{2} i^{2} \left (d a -b c \right )^{3}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b}{g^{2} i^{2} \left (d a -b c \right )^{3}}-\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (d x +c \right )}+\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b c}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (d x +c \right )}+\frac {B \,d^{2} a}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (d x +c \right )}-\frac {B d b c}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (d x +c \right )}+\frac {B b d}{g^{2} i^{2} \left (d a -b c \right )^{3}}+\frac {B b d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i^{2} \left (d a -b c \right )^{3}}+\frac {B e \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {B e \,b^{2}}{g^{2} i^{2} \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}\) | \(541\) |
| norman | \(\frac {\frac {\left (2 A a b c d -B \,a^{2} d^{2}+B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A a b \,d^{2}+2 A \,b^{2} c d -2 B a b \,d^{2}+2 B \,b^{2} c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} B \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {2 d^{2} b^{2} A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B a b c d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (d a +b c \right ) B b d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {A a b \,d^{2}+A \,b^{2} c d -B a b \,d^{2}+B \,b^{2} c d}{i g \left (d a -b c \right )^{2} d b}-\frac {2 A b d x}{i g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}}{g \left (b x +a \right ) i \left (d x +c \right )}\) | \(574\) |
| parallelrisch | \(\frac {2 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b \,c^{3} d^{2}+2 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} c^{4} d -2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b \,c^{3} d^{2}+2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} c^{4} d +A x \,a^{5} c^{2} d^{3}-A x \,a^{2} b^{3} c^{5}-B x \,a^{5} c^{2} d^{3}-B x \,a^{2} b^{3} c^{5}-B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} c^{3} d^{2}+B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} c^{5}-B \,x^{2} a^{4} b \,c^{2} d^{3}+2 B \,x^{2} a^{3} b^{2} c^{3} d^{2}-B \,x^{2} a^{2} b^{3} c^{4} d +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} b \,c^{4} d -A x \,a^{4} b \,c^{3} d^{2}+A x \,a^{3} b^{2} c^{4} d +2 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b \,c^{4} d +B x \,a^{4} b \,c^{3} d^{2}+B x \,a^{3} b^{2} c^{4} d +B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b^{2} c^{3} d^{2}+A \,x^{2} a^{4} b \,c^{2} d^{3}-A \,x^{2} a^{2} b^{3} c^{4} d +2 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} c^{3} d^{2}+B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} b \,c^{3} d^{2}+B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b^{2} c^{4} d}{i^{2} g^{2} \left (d x +c \right ) \left (b x +a \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{3} c^{3}}\) | \(601\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x,method=_RETU RNVERBOSE)
Output:
1/g^2*A/i^2*(-d/(a*d-b*c)^2/(d*x+c)-2*d/(a*d-b*c)^3*b*ln(d*x+c)-b/(a*d-b*c )^2/(b*x+a)+2*d/(a*d-b*c)^3*b*ln(b*x+a))-B/g^2/i^2/(a*d-b*c)/e*(d^2/(a*d-b *c)^2*((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)-1/(a*d-b*c)^2*b*d*e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c) )^2+1/(a*d-b*c)^2*b^2*e^2*(-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 = 0.08 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.28 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {{\left (A + B\right )} b^{2} c^{2} - 2 \, B a b c d - {\left (A - B\right )} a^{2} d^{2} + {\left (B b^{2} d^{2} x^{2} + B a b c d + {\left (B b^{2} c d + B a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A b^{2} c d - A a b d^{2}\right )} x + {\left (2 \, A b^{2} d^{2} x^{2} + B b^{2} c^{2} + 2 \, A a b c d - B a^{2} d^{2} + 2 \, {\left ({\left (A + B\right )} b^{2} c d + {\left (A - B\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algo rithm="fricas")
Output:
-((A + B)*b^2*c^2 - 2*B*a*b*c*d - (A - B)*a^2*d^2 + (B*b^2*d^2*x^2 + B*a*b *c*d + (B*b^2*c*d + B*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*b^ 2*c*d - A*a*b*d^2)*x + (2*A*b^2*d^2*x^2 + B*b^2*c^2 + 2*A*a*b*c*d - B*a^2* d^2 + 2*((A + B)*b^2*c*d + (A - B)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c) ))/((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*g^2*i^2*x^ 2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*g^2*i^2*x + (a*b^3 *c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*g^2*i^2)
Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (235) = 470\).
Time = 1.88 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.17 \[ \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)^2} \, dx=- \frac {2 A b d \log {\left (x + \frac {- \frac {2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac {8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} - \frac {2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac {2 A b d \log {\left (x + \frac {\frac {2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac {8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} + \frac {2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac {B b d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{3} g^{2} i^{2} - 3 a^{2} b c d^{2} g^{2} i^{2} + 3 a b^{2} c^{2} d g^{2} i^{2} - b^{3} c^{3} g^{2} i^{2}} + \frac {\left (- B a d - B b c - 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{3} c d^{2} g^{2} i^{2} + a^{3} d^{3} g^{2} i^{2} x - 2 a^{2} b c^{2} d g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} x + a^{2} b d^{3} g^{2} i^{2} x^{2} + a b^{2} c^{3} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} x - 2 a b^{2} c d^{2} g^{2} i^{2} x^{2} + b^{3} c^{3} g^{2} i^{2} x + b^{3} c^{2} d g^{2} i^{2} x^{2}} - \frac {A a d + A b c + 2 A b d x - B a d + B b c}{a^{3} c d^{2} g^{2} i^{2} - 2 a^{2} b c^{2} d g^{2} i^{2} + a b^{2} c^{3} g^{2} i^{2} + x^{2} \left (a^{2} b d^{3} g^{2} i^{2} - 2 a b^{2} c d^{2} g^{2} i^{2} + b^{3} c^{2} d g^{2} i^{2}\right ) + x \left (a^{3} d^{3} g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} + b^{3} c^{3} g^{2} i^{2}\right )} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**2/(d*i*x+c*i)**2,x)
Output:
-2*A*b*d*log(x + (-2*A*a**4*b*d**5/(a*d - b*c)**3 + 8*A*a**3*b**2*c*d**4/( a*d - b*c)**3 - 12*A*a**2*b**3*c**2*d**3/(a*d - b*c)**3 + 8*A*a*b**4*c**3* d**2/(a*d - b*c)**3 + 2*A*a*b*d**2 - 2*A*b**5*c**4*d/(a*d - b*c)**3 + 2*A* b**2*c*d)/(4*A*b**2*d**2))/(g**2*i**2*(a*d - b*c)**3) + 2*A*b*d*log(x + (2 *A*a**4*b*d**5/(a*d - b*c)**3 - 8*A*a**3*b**2*c*d**4/(a*d - b*c)**3 + 12*A *a**2*b**3*c**2*d**3/(a*d - b*c)**3 - 8*A*a*b**4*c**3*d**2/(a*d - b*c)**3 + 2*A*a*b*d**2 + 2*A*b**5*c**4*d/(a*d - b*c)**3 + 2*A*b**2*c*d)/(4*A*b**2* d**2))/(g**2*i**2*(a*d - b*c)**3) + B*b*d*log(e*(a + b*x)/(c + d*x))**2/(a **3*d**3*g**2*i**2 - 3*a**2*b*c*d**2*g**2*i**2 + 3*a*b**2*c**2*d*g**2*i**2 - b**3*c**3*g**2*i**2) + (-B*a*d - B*b*c - 2*B*b*d*x)*log(e*(a + b*x)/(c + d*x))/(a**3*c*d**2*g**2*i**2 + a**3*d**3*g**2*i**2*x - 2*a**2*b*c**2*d*g **2*i**2 - a**2*b*c*d**2*g**2*i**2*x + a**2*b*d**3*g**2*i**2*x**2 + a*b**2 *c**3*g**2*i**2 - a*b**2*c**2*d*g**2*i**2*x - 2*a*b**2*c*d**2*g**2*i**2*x* *2 + b**3*c**3*g**2*i**2*x + b**3*c**2*d*g**2*i**2*x**2) - (A*a*d + A*b*c + 2*A*b*d*x - B*a*d + B*b*c)/(a**3*c*d**2*g**2*i**2 - 2*a**2*b*c**2*d*g**2 *i**2 + a*b**2*c**3*g**2*i**2 + x**2*(a**2*b*d**3*g**2*i**2 - 2*a*b**2*c*d **2*g**2*i**2 + b**3*c**2*d*g**2*i**2) + x*(a**3*d**3*g**2*i**2 - a**2*b*c *d**2*g**2*i**2 - a*b**2*c**2*d*g**2*i**2 + b**3*c**3*g**2*i**2))
Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (261) = 522\).
Time = 0.08 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.29 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algo rithm="maxima")
Output:
-B*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*g^2*i^2 *x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*g^2*i^2*x + (a*b^2* c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2) - 2*b*d*log(d*x + c)/(( b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2))*log(b*e*x/(d* x + c) + a*e/(d*x + c)) - A*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c *d^2 + a^2*b*d^3)*g^2*i^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3 *d^3)*g^2*i^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^2) + 2*b*d *log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2 ) - 2*b*d*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 )*g^2*i^2)) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2 - (b^2*d^2*x^2 + a*b*c*d + (b ^2*c*d + a*b*d^2)*x)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*log(d*x + c) - (b^2*d^2*x^2 + a*b*c*d + (b^2*c* d + a*b*d^2)*x)*log(d*x + c)^2)*B/(a*b^3*c^4*g^2*i^2 - 3*a^2*b^2*c^3*d*g^2 *i^2 + 3*a^3*b*c^2*d^2*g^2*i^2 - a^4*c*d^3*g^2*i^2 + (b^4*c^3*d*g^2*i^2 - 3*a*b^3*c^2*d^2*g^2*i^2 + 3*a^2*b^2*c*d^3*g^2*i^2 - a^3*b*d^4*g^2*i^2)*x^2 + (b^4*c^4*g^2*i^2 - 2*a*b^3*c^3*d*g^2*i^2 + 2*a^3*b*c*d^3*g^2*i^2 - a^4* d^4*g^2*i^2)*x)
Time = 42.64 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.51 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-{\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} {\left (\frac {{\left (d x + c\right )} B e^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )} g^{2} i^{2}} + \frac {{\left (A e^{2} + B e^{2}\right )} {\left (d x + c\right )}}{{\left (b e x + a e\right )} g^{2} i^{2}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algo rithm="giac")
Output:
-(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2 *((d*x + c)*B*e^2*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)*g^2*i^2) + ( A*e^2 + B*e^2)*(d*x + c)/((b*e*x + a*e)*g^2*i^2))
Time = 27.99 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.59 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {B\,b\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}+\frac {B\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,A\,b\,d\,x}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,4{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^2*(c*i + d*i*x)^2) ,x)
Output:
(B*b*d*log((e*(a + b*x))/(c + d*x))^2)/(g^2*i^2*(a*d - b*c)^3) - (A*b*d*at an((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*4i)/(g^2*i^2*(a*d - b*c)^3) - (A*a*d)/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x)) - (A*b*c)/(g^2*i^2*(a *d - b*c)^2*(a + b*x)*(c + d*x)) + (B*a*d)/(g^2*i^2*(a*d - b*c)^2*(a + b*x )*(c + d*x)) - (B*b*c)/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x)) - (2*A* b*d*x)/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x)) - (B*a*d*log((e*(a + b* x))/(c + d*x)))/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x)) - (B*b*c*log(( e*(a + b*x))/(c + d*x)))/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x)) - (2* B*b*d*x*log((e*(a + b*x))/(c + d*x)))/(g^2*i^2*(a*d - b*c)^2*(a + b*x)*(c + d*x))
Time = 0.20 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.43 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x)
Output:
( - 2*log(a + b*x)*a**3*b*c*d**2 - 2*log(a + b*x)*a**3*b*d**3*x - 2*log(a + b*x)*a**2*b**2*c**2*d - 4*log(a + b*x)*a**2*b**2*c*d**2*x - 2*log(a + b* x)*a**2*b**2*d**3*x**2 - 2*log(a + b*x)*a*b**3*c**2*d*x - 2*log(a + b*x)*a *b**3*c*d**2*x**2 + 2*log(c + d*x)*a**3*b*c*d**2 + 2*log(c + d*x)*a**3*b*d **3*x + 2*log(c + d*x)*a**2*b**2*c**2*d + 4*log(c + d*x)*a**2*b**2*c*d**2* x + 2*log(c + d*x)*a**2*b**2*d**3*x**2 + 2*log(c + d*x)*a*b**3*c**2*d*x + 2*log(c + d*x)*a*b**3*c*d**2*x**2 - log((a*e + b*e*x)/(c + d*x))**2*a**2*b **2*c*d**2 - log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*d**3*x - log((a*e + b*e*x)/(c + d*x))**2*a*b**3*c**2*d - 2*log((a*e + b*e*x)/(c + d*x))**2*a* b**3*c*d**2*x - log((a*e + b*e*x)/(c + d*x))**2*a*b**3*d**3*x**2 - log((a* e + b*e*x)/(c + d*x))**2*b**4*c**2*d*x - log((a*e + b*e*x)/(c + d*x))**2*b **4*c*d**2*x**2 + log((a*e + b*e*x)/(c + d*x))*a**3*b*d**3 + log((a*e + b* e*x)/(c + d*x))*a**2*b**2*c*d**2 + 2*log((a*e + b*e*x)/(c + d*x))*a**2*b** 2*d**3*x - log((a*e + b*e*x)/(c + d*x))*a*b**3*c**2*d - log((a*e + b*e*x)/ (c + d*x))*b**4*c**3 - 2*log((a*e + b*e*x)/(c + d*x))*b**4*c**2*d*x + a**4 *d**3 - a**3*b*c*d**2 - a**3*b*d**3 + a**2*b**2*c**2*d + a**2*b**2*c*d**2 - 2*a**2*b**2*d**3*x**2 - a*b**3*c**3 + a*b**3*c**2*d + 2*a*b**3*c*d**2*x* *2 - b**4*c**3)/(g**2*(a**5*c*d**4 + a**5*d**5*x - 2*a**4*b*c**2*d**3 - a* *4*b*c*d**4*x + a**4*b*d**5*x**2 - 2*a**3*b**2*c**2*d**3*x - 2*a**3*b**2*c *d**4*x**2 + 2*a**2*b**3*c**4*d + 2*a**2*b**3*c**3*d**2*x - a*b**4*c**5...