Integrand size = 40, antiderivative size = 281 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {B d^2 i^2 (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \]
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Time = 0.14 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2372, 12, 14} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i^2 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {B d^2 i^2 (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B d i^2 (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]
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Rule 12
Rule 14
Rule 45
Rule 2372
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {i^2 \text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{30 x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \left (-\frac {6 b^2}{x^6}+\frac {15 b d}{x^5}-\frac {10 d^2}{x^4}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {B d^2 i^2 (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.22 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\frac {i^2 \left (-\frac {360 A b^2 c^2}{(a+b x)^5}-\frac {72 b^2 B c^2}{(a+b x)^5}+\frac {720 a A b c d}{(a+b x)^5}+\frac {144 a b B c d}{(a+b x)^5}-\frac {360 a^2 A d^2}{(a+b x)^5}-\frac {72 a^2 B d^2}{(a+b x)^5}-\frac {900 A b c d}{(a+b x)^4}-\frac {135 b B c d}{(a+b x)^4}+\frac {900 a A d^2}{(a+b x)^4}+\frac {135 a B d^2}{(a+b x)^4}-\frac {600 A d^2}{(a+b x)^3}-\frac {20 B d^2}{(a+b x)^3}+\frac {30 B d^3}{(b c-a d) (a+b x)^2}-\frac {60 B d^4}{(b c-a d)^2 (a+b x)}-\frac {60 B d^5 \log (a+b x)}{(b c-a d)^3}-\frac {60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5}+\frac {60 B d^5 \log (c+d x)}{(b c-a d)^3}\right )}{1800 b^3 g^6} \]
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Time = 1.70 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.53
method | result | size |
parts | \(\frac {i^{2} A \left (-\frac {d^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {d \left (a d -c b \right )}{2 b^{3} \left (b x +a \right )^{4}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{5 b^{3} \left (b x +a \right )^{5}}\right )}{g^{6}}-\frac {i^{2} B \left (a d -c b \right )^{3} e^{3} \left (\frac {d^{6} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{6}}-\frac {2 d^{5} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{6}}+\frac {d^{4} e^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{6}}\right )}{g^{6} d^{4}}\) | \(430\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{4} A \,b^{2}}{5 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}+\frac {i^{2} d^{3} e^{3} A b}{2 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{4} e^{2} A}{3 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{4} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{4} g^{6}}-\frac {2 i^{2} d^{3} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{4} g^{6}}+\frac {i^{2} d^{4} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4} g^{6}}\right )}{d^{2}}\) | \(532\) |
default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{4} A \,b^{2}}{5 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}+\frac {i^{2} d^{3} e^{3} A b}{2 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{4} e^{2} A}{3 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{4} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{4} g^{6}}-\frac {2 i^{2} d^{3} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{4} g^{6}}+\frac {i^{2} d^{4} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4} g^{6}}\right )}{d^{2}}\) | \(532\) |
risch | \(-\frac {i^{2} B \left (10 d^{2} x^{2} b^{2}+5 a b \,d^{2} x +15 b^{2} c d x +a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{30 \left (b x +a \right )^{5} g^{6} b^{3}}-\frac {\left (60 B \ln \left (d x +c \right ) a^{5} d^{5}-72 B \,b^{5} c^{5}+60 B a \,b^{4} d^{5} x^{4}-60 B \,b^{5} c \,d^{4} x^{4}+270 B \,a^{2} b^{3} d^{5} x^{3}+30 B \,b^{5} c^{2} d^{3} x^{3}+600 A \,a^{3} b^{2} d^{5} x^{2}+470 B \,a^{3} b^{2} d^{5} x^{2}-20 B \,b^{5} c^{3} d^{2} x^{2}+300 A \,a^{4} b \,d^{5} x -1800 A \,a^{2} b^{3} c \,d^{4} x^{2}-300 B a \,b^{4} c \,d^{4} x^{3}-600 B \,a^{2} b^{3} c \,d^{4} x^{2}+150 B a \,b^{4} c^{2} d^{3} x^{2}-600 B \,a^{2} b^{3} c^{2} d^{3} x +500 B a \,b^{4} c^{3} d^{2} x -60 B \ln \left (-b x -a \right ) a^{5} d^{5}+235 B \,a^{4} b \,d^{5} x -135 B \,b^{5} c^{4} d x +1800 A a \,b^{4} c^{2} d^{3} x^{2}-1800 A \,a^{2} b^{3} c^{2} d^{3} x +2400 A a \,b^{4} c^{3} d^{2} x -300 B \ln \left (-b x -a \right ) a \,b^{4} d^{5} x^{4}+300 B \ln \left (d x +c \right ) a \,b^{4} d^{5} x^{4}-600 B \ln \left (-b x -a \right ) a^{2} b^{3} d^{5} x^{3}+600 B \ln \left (d x +c \right ) a^{2} b^{3} d^{5} x^{3}-600 B \ln \left (-b x -a \right ) a^{3} b^{2} d^{5} x^{2}+60 A \,a^{5} d^{5}+600 B \ln \left (d x +c \right ) a^{3} b^{2} d^{5} x^{2}-300 B \ln \left (-b x -a \right ) a^{4} b \,d^{5} x +300 B \ln \left (d x +c \right ) a^{4} b \,d^{5} x -360 A \,b^{5} c^{5}+47 B \,a^{5} d^{5}-600 A \,a^{2} b^{3} c^{3} d^{2}+900 A a \,b^{4} c^{4} d -200 B \,a^{2} b^{3} c^{3} d^{2}+225 B a \,b^{4} c^{4} d -600 A \,b^{5} c^{3} d^{2} x^{2}-900 A \,b^{5} c^{4} d x -60 B \ln \left (-b x -a \right ) b^{5} d^{5} x^{5}+60 B \ln \left (d x +c \right ) b^{5} d^{5} x^{5}\right ) i^{2}}{1800 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{5} g^{6} b^{3}}\) | \(768\) |
parallelrisch | \(-\frac {-1800 A \,x^{2} a^{2} b^{7} c \,d^{5} i^{2}+1800 A \,x^{2} a \,b^{8} c^{2} d^{4} i^{2}-600 B \,x^{2} a^{2} b^{7} c \,d^{5} i^{2}-1800 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c^{2} d^{4} i^{2}-300 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} d^{6} i^{2}-600 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} d^{6} i^{2}-300 B \,x^{3} a \,b^{8} c \,d^{5} i^{2}-600 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{3} d^{3} i^{2}+150 B \,x^{2} a \,b^{8} c^{2} d^{4} i^{2}-900 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{4} d^{2} i^{2}-1800 A x \,a^{2} b^{7} c^{2} d^{4} i^{2}+2400 A x a \,b^{8} c^{3} d^{3} i^{2}-600 B x \,a^{2} b^{7} c^{2} d^{4} i^{2}+500 B x a \,b^{8} c^{3} d^{3} i^{2}-600 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c^{3} d^{3} i^{2}+900 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{4} d^{2} i^{2}-200 B \,a^{2} b^{7} c^{3} d^{3} i^{2}+225 B a \,b^{8} c^{4} d^{2} i^{2}-60 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} d^{6} i^{2}+60 B \,x^{4} a \,b^{8} d^{6} i^{2}-60 B \,x^{4} b^{9} c \,d^{5} i^{2}+270 B \,x^{3} a^{2} b^{7} d^{6} i^{2}+30 B \,x^{3} b^{9} c^{2} d^{4} i^{2}+600 A \,x^{2} a^{3} b^{6} d^{6} i^{2}-600 A \,x^{2} b^{9} c^{3} d^{3} i^{2}+470 B \,x^{2} a^{3} b^{6} d^{6} i^{2}-20 B \,x^{2} b^{9} c^{3} d^{3} i^{2}+300 A x \,a^{4} b^{5} d^{6} i^{2}-900 A x \,b^{9} c^{4} d^{2} i^{2}+235 B x \,a^{4} b^{5} d^{6} i^{2}-135 B x \,b^{9} c^{4} d^{2} i^{2}-360 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{5} d \,i^{2}+2400 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{3} d^{3} i^{2}-600 A \,a^{2} b^{7} c^{3} d^{3} i^{2}+900 A a \,b^{8} c^{4} d^{2} i^{2}-1800 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c \,d^{5} i^{2}+1800 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{2} d^{4} i^{2}+60 A \,a^{5} b^{4} d^{6} i^{2}-360 A \,b^{9} c^{5} d \,i^{2}+47 B \,a^{5} b^{4} d^{6} i^{2}-72 B \,b^{9} c^{5} d \,i^{2}}{1800 g^{6} \left (b x +a \right )^{5} \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^{7} d}\) | \(935\) |
norman | \(\text {Expression too large to display}\) | \(1043\) |
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Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (269) = 538\).
Time = 0.37 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.87 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {60 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{2} x^{4} - 30 \, {\left (B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} + 9 \, B a^{2} b^{3} d^{5}\right )} i^{2} x^{3} + 10 \, {\left (2 \, {\left (30 \, A + B\right )} b^{5} c^{3} d^{2} - 15 \, {\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 60 \, {\left (3 \, A + B\right )} a^{2} b^{3} c d^{4} - {\left (60 \, A + 47 \, B\right )} a^{3} b^{2} d^{5}\right )} i^{2} x^{2} + 5 \, {\left (9 \, {\left (20 \, A + 3 \, B\right )} b^{5} c^{4} d - 20 \, {\left (24 \, A + 5 \, B\right )} a b^{4} c^{3} d^{2} + 120 \, {\left (3 \, A + B\right )} a^{2} b^{3} c^{2} d^{3} - {\left (60 \, A + 47 \, B\right )} a^{4} b d^{5}\right )} i^{2} x + {\left (72 \, {\left (5 \, A + B\right )} b^{5} c^{5} - 225 \, {\left (4 \, A + B\right )} a b^{4} c^{4} d + 200 \, {\left (3 \, A + B\right )} a^{2} b^{3} c^{3} d^{2} - {\left (60 \, A + 47 \, B\right )} a^{5} d^{5}\right )} i^{2} + 60 \, {\left (B b^{5} d^{5} i^{2} x^{5} + 5 \, B a b^{4} d^{5} i^{2} x^{4} + 10 \, B a^{2} b^{3} d^{5} i^{2} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4}\right )} i^{2} x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3}\right )} i^{2} x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{1800 \, {\left ({\left (b^{11} c^{3} - 3 \, a b^{10} c^{2} d + 3 \, a^{2} b^{9} c d^{2} - a^{3} b^{8} d^{3}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{3} - 3 \, a^{2} b^{9} c^{2} d + 3 \, a^{3} b^{8} c d^{2} - a^{4} b^{7} d^{3}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{3} - 3 \, a^{3} b^{8} c^{2} d + 3 \, a^{4} b^{7} c d^{2} - a^{5} b^{6} d^{3}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{3} - 3 \, a^{4} b^{7} c^{2} d + 3 \, a^{5} b^{6} c d^{2} - a^{6} b^{5} d^{3}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{3} - 3 \, a^{5} b^{6} c^{2} d + 3 \, a^{6} b^{5} c d^{2} - a^{7} b^{4} d^{3}\right )} g^{6} x + {\left (a^{5} b^{6} c^{3} - 3 \, a^{6} b^{5} c^{2} d + 3 \, a^{7} b^{4} c d^{2} - a^{8} b^{3} d^{3}\right )} g^{6}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1300 vs. \(2 (258) = 516\).
Time = 76.59 (sec) , antiderivative size = 1300, normalized size of antiderivative = 4.63 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=- \frac {B d^{5} i^{2} \log {\left (x + \frac {- \frac {B a^{4} d^{9} i^{2}}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{8} i^{2}}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{7} i^{2}}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{6} i^{2}}{\left (a d - b c\right )^{3}} + B a d^{6} i^{2} - \frac {B b^{4} c^{4} d^{5} i^{2}}{\left (a d - b c\right )^{3}} + B b c d^{5} i^{2}}{2 B b d^{6} i^{2}} \right )}}{30 b^{3} g^{6} \left (a d - b c\right )^{3}} + \frac {B d^{5} i^{2} \log {\left (x + \frac {\frac {B a^{4} d^{9} i^{2}}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{8} i^{2}}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{7} i^{2}}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{6} i^{2}}{\left (a d - b c\right )^{3}} + B a d^{6} i^{2} + \frac {B b^{4} c^{4} d^{5} i^{2}}{\left (a d - b c\right )^{3}} + B b c d^{5} i^{2}}{2 B b d^{6} i^{2}} \right )}}{30 b^{3} g^{6} \left (a d - b c\right )^{3}} + \frac {- 60 A a^{4} d^{4} i^{2} - 60 A a^{3} b c d^{3} i^{2} - 60 A a^{2} b^{2} c^{2} d^{2} i^{2} + 540 A a b^{3} c^{3} d i^{2} - 360 A b^{4} c^{4} i^{2} - 47 B a^{4} d^{4} i^{2} - 47 B a^{3} b c d^{3} i^{2} - 47 B a^{2} b^{2} c^{2} d^{2} i^{2} + 153 B a b^{3} c^{3} d i^{2} - 72 B b^{4} c^{4} i^{2} - 60 B b^{4} d^{4} i^{2} x^{4} + x^{3} \left (- 270 B a b^{3} d^{4} i^{2} + 30 B b^{4} c d^{3} i^{2}\right ) + x^{2} \left (- 600 A a^{2} b^{2} d^{4} i^{2} + 1200 A a b^{3} c d^{3} i^{2} - 600 A b^{4} c^{2} d^{2} i^{2} - 470 B a^{2} b^{2} d^{4} i^{2} + 130 B a b^{3} c d^{3} i^{2} - 20 B b^{4} c^{2} d^{2} i^{2}\right ) + x \left (- 300 A a^{3} b d^{4} i^{2} - 300 A a^{2} b^{2} c d^{3} i^{2} + 1500 A a b^{3} c^{2} d^{2} i^{2} - 900 A b^{4} c^{3} d i^{2} - 235 B a^{3} b d^{4} i^{2} - 235 B a^{2} b^{2} c d^{3} i^{2} + 365 B a b^{3} c^{2} d^{2} i^{2} - 135 B b^{4} c^{3} d i^{2}\right )}{1800 a^{7} b^{3} d^{2} g^{6} - 3600 a^{6} b^{4} c d g^{6} + 1800 a^{5} b^{5} c^{2} g^{6} + x^{5} \cdot \left (1800 a^{2} b^{8} d^{2} g^{6} - 3600 a b^{9} c d g^{6} + 1800 b^{10} c^{2} g^{6}\right ) + x^{4} \cdot \left (9000 a^{3} b^{7} d^{2} g^{6} - 18000 a^{2} b^{8} c d g^{6} + 9000 a b^{9} c^{2} g^{6}\right ) + x^{3} \cdot \left (18000 a^{4} b^{6} d^{2} g^{6} - 36000 a^{3} b^{7} c d g^{6} + 18000 a^{2} b^{8} c^{2} g^{6}\right ) + x^{2} \cdot \left (18000 a^{5} b^{5} d^{2} g^{6} - 36000 a^{4} b^{6} c d g^{6} + 18000 a^{3} b^{7} c^{2} g^{6}\right ) + x \left (9000 a^{6} b^{4} d^{2} g^{6} - 18000 a^{5} b^{5} c d g^{6} + 9000 a^{4} b^{6} c^{2} g^{6}\right )} + \frac {\left (- B a^{2} d^{2} i^{2} - 3 B a b c d i^{2} - 5 B a b d^{2} i^{2} x - 6 B b^{2} c^{2} i^{2} - 15 B b^{2} c d i^{2} x - 10 B b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{30 a^{5} b^{3} g^{6} + 150 a^{4} b^{4} g^{6} x + 300 a^{3} b^{5} g^{6} x^{2} + 300 a^{2} b^{6} g^{6} x^{3} + 150 a b^{7} g^{6} x^{4} + 30 b^{8} g^{6} x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3029 vs. \(2 (269) = 538\).
Time = 0.35 (sec) , antiderivative size = 3029, normalized size of antiderivative = 10.78 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\text {Too large to display} \]
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Time = 0.55 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {1}{1800} \, {\left (\frac {60 \, {\left (6 \, B b^{2} e^{6} i^{2} - \frac {15 \, {\left (b e x + a e\right )} B b d e^{5} i^{2}}{d x + c} + \frac {10 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b e x + a e\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b e x + a e\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {360 \, A b^{2} e^{6} i^{2} + 72 \, B b^{2} e^{6} i^{2} - \frac {900 \, {\left (b e x + a e\right )} A b d e^{5} i^{2}}{d x + c} - \frac {225 \, {\left (b e x + a e\right )} B b d e^{5} i^{2}}{d x + c} + \frac {600 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}} + \frac {200 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b e x + a e\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b e x + a e\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
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Time = 4.61 (sec) , antiderivative size = 941, normalized size of antiderivative = 3.35 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\frac {B\,d^5\,i^2\,\mathrm {atanh}\left (\frac {30\,a^3\,b^3\,d^3\,g^6-30\,a^2\,b^4\,c\,d^2\,g^6-30\,a\,b^5\,c^2\,d\,g^6+30\,b^6\,c^3\,g^6}{30\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{15\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^4\,g^6}+\frac {B\,c\,d\,i^2}{10\,b^3\,g^6}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^4\,g^6}+\frac {B\,c\,d\,i^2}{10\,b^3\,g^6}\right )+\frac {2\,B\,a\,d^2\,i^2}{15\,b^3\,g^6}+\frac {2\,B\,c\,d\,i^2}{5\,b^2\,g^6}\right )+\frac {B\,c^2\,i^2}{5\,b^2\,g^6}+\frac {B\,d^2\,i^2\,x^2}{3\,b^2\,g^6}\right )}{5\,a^4\,x+\frac {a^5}{b}+b^4\,x^5+10\,a^3\,b\,x^2+5\,a\,b^3\,x^4+10\,a^2\,b^2\,x^3}-\frac {\frac {60\,A\,a^4\,d^4\,i^2+360\,A\,b^4\,c^4\,i^2+47\,B\,a^4\,d^4\,i^2+72\,B\,b^4\,c^4\,i^2+60\,A\,a^2\,b^2\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,c^2\,d^2\,i^2-540\,A\,a\,b^3\,c^3\,d\,i^2+60\,A\,a^3\,b\,c\,d^3\,i^2-153\,B\,a\,b^3\,c^3\,d\,i^2+47\,B\,a^3\,b\,c\,d^3\,i^2}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^2\,b^2\,d^4\,i^2+47\,B\,a^2\,b^2\,d^4\,i^2+60\,A\,b^4\,c^2\,d^2\,i^2+2\,B\,b^4\,c^2\,d^2\,i^2-120\,A\,a\,b^3\,c\,d^3\,i^2-13\,B\,a\,b^3\,c\,d^3\,i^2\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^3\,b\,d^4\,i^2+47\,B\,a^3\,b\,d^4\,i^2+180\,A\,b^4\,c^3\,d\,i^2+27\,B\,b^4\,c^3\,d\,i^2-300\,A\,a\,b^3\,c^2\,d^2\,i^2+60\,A\,a^2\,b^2\,c\,d^3\,i^2-73\,B\,a\,b^3\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,c\,d^3\,i^2\right )}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^3\,\left (9\,B\,a\,b^3\,d^3\,i^2-B\,b^4\,c\,d^2\,i^2\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^4\,d^4\,i^2\,x^4}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{30\,a^5\,b^3\,g^6+150\,a^4\,b^4\,g^6\,x+300\,a^3\,b^5\,g^6\,x^2+300\,a^2\,b^6\,g^6\,x^3+150\,a\,b^7\,g^6\,x^4+30\,b^8\,g^6\,x^5} \]
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