Integrand size = 30, antiderivative size = 180 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^4 x}{5 d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2}{10 b d^3}-\frac {B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4}{20 b d}+\frac {B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b} \]
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Time = 0.08 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2548, 21, 45} \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b}+\frac {B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac {B g^4 x (b c-a d)^4}{5 d^4}+\frac {B g^4 (a+b x)^2 (b c-a d)^3}{10 b d^3}-\frac {B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}+\frac {B g^4 (a+b x)^4 (b c-a d)}{20 b d} \]
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Rule 21
Rule 45
Rule 2548
Rubi steps \begin{align*} \text {integral}& = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b}+\frac {(B (b c-a d)) \int \frac {(a g+b g x)^5}{(a+b x) (c+d x)} \, dx}{5 b g} \\ & = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b}+\frac {\left (B (b c-a d) g^4\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b} \\ & = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b}+\frac {\left (B (b c-a d) g^4\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b} \\ & = -\frac {B (b c-a d)^4 g^4 x}{5 d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2}{10 b d^3}-\frac {B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4}{20 b d}+\frac {B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.79 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {g^4 \left (-\frac {B (-b c+a d) \left (-12 b d (b c-a d)^3 x+6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (-b c+a d) (a+b x)^3+3 d^4 (a+b x)^4+12 (b c-a d)^4 \log (c+d x)\right )}{12 d^5}+(a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )\right )}{5 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(168)=336\).
Time = 1.16 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.46
method | result | size |
risch | \(\frac {g^{4} b^{4} A \,x^{5}}{5}+g^{4} b^{3} A a \,x^{4}-\frac {g^{4} b^{3} B a \,x^{4}}{20}+\frac {g^{4} b^{4} B c \,x^{4}}{20 d}+2 g^{4} b^{2} A \,a^{2} x^{3}-\frac {4 g^{4} b^{2} B \,a^{2} x^{3}}{15}-\frac {g^{4} b^{4} B \,c^{2} x^{3}}{15 d^{2}}+2 g^{4} b A \,a^{3} x^{2}-\frac {3 g^{4} b B \,a^{3} x^{2}}{5}+\frac {g^{4} b^{4} B \,c^{3} x^{2}}{10 d^{3}}+g^{4} A \,a^{4} x -\frac {4 g^{4} B \,a^{4} x}{5}-\frac {g^{4} b^{4} B \,c^{4} x}{5 d^{4}}+\frac {g^{4} b^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}+\frac {g^{4} B \ln \left (d x +c \right ) a^{4} c}{d}+\frac {2 g^{4} b B \,a^{3} c x}{d}-\frac {2 g^{4} b^{2} B \,a^{2} c^{2} x}{d^{2}}+\frac {g^{4} b^{3} B a \,c^{3} x}{d^{3}}-\frac {2 g^{4} b B \ln \left (d x +c \right ) a^{3} c^{2}}{d^{2}}+\frac {2 g^{4} b^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{d^{3}}-\frac {g^{4} b^{3} B \ln \left (d x +c \right ) a \,c^{4}}{d^{4}}+\frac {g^{4} b^{3} B a c \,x^{3}}{3 d}-\frac {g^{4} B \ln \left (d x +c \right ) a^{5}}{5 b}+\frac {g^{4} b^{2} B \,a^{2} c \,x^{2}}{d}-\frac {g^{4} b^{3} B a \,c^{2} x^{2}}{2 d^{2}}+\frac {\left (b x +a \right )^{5} g^{4} B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{5 b}\) | \(442\) |
parts | \(\frac {A \,g^{4} \left (b x +a \right )^{5}}{5 b}+B \,g^{4} e^{5} \left (a d -c b \right )^{5} \left (-\frac {1}{20 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{4}}+\frac {1}{15 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{5 d^{5} e^{5} b}+\frac {1}{5 d^{4} e^{4} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {1}{10 d^{3} e^{3} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{4} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}-5 b^{3} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} d e +10 b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} d^{2} e^{2}-10 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b \,d^{3} e^{3}+5 d^{4} e^{4}\right )}{5 d^{5} e^{5} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{5}}\right )\) | \(536\) |
derivativedivides | \(\frac {e \left (a d -c b \right ) \left (-\frac {A b \,e^{4} g^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{5 \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{5}}+B \,b^{2} e^{4} g^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\frac {1}{20 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{4}}+\frac {1}{15 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{5 d^{5} e^{5} b}+\frac {1}{5 d^{4} e^{4} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {1}{10 d^{3} e^{3} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{4} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}-5 b^{3} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} d e +10 b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} d^{2} e^{2}-10 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b \,d^{3} e^{3}+5 d^{4} e^{4}\right )}{5 d^{5} e^{5} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{5}}\right )\right )}{b^{2}}\) | \(671\) |
default | \(\frac {e \left (a d -c b \right ) \left (-\frac {A b \,e^{4} g^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{5 \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{5}}+B \,b^{2} e^{4} g^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\frac {1}{20 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{4}}+\frac {1}{15 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{5 d^{5} e^{5} b}+\frac {1}{5 d^{4} e^{4} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {1}{10 d^{3} e^{3} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{4} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}-5 b^{3} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} d e +10 b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} d^{2} e^{2}-10 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b \,d^{3} e^{3}+5 d^{4} e^{4}\right )}{5 d^{5} e^{5} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{5}}\right )\right )}{b^{2}}\) | \(671\) |
parallelrisch | \(\frac {120 B x \,a^{3} b^{2} c \,d^{4} g^{4}-120 B x \,a^{2} b^{3} c^{2} d^{3} g^{4}+60 B x a \,b^{4} c^{3} d^{2} g^{4}+60 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{4} b c \,d^{4} g^{4}-120 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{3} b^{2} c^{2} d^{3} g^{4}+120 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{3} c^{3} d^{2} g^{4}-60 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{4} c^{4} d \,g^{4}+60 B \ln \left (b x +a \right ) a^{4} b c \,d^{4} g^{4}-120 B \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d^{3} g^{4}+120 B \ln \left (b x +a \right ) a^{2} b^{3} c^{3} d^{2} g^{4}-60 B \ln \left (b x +a \right ) a \,b^{4} c^{4} d \,g^{4}+60 B \,x^{4} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{4} d^{5} g^{4}-12 B x \,b^{5} c^{4} d \,g^{4}+12 B \,x^{5} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{5} d^{5} g^{4}+60 A \,x^{4} a \,b^{4} d^{5} g^{4}-3 B \,x^{4} a \,b^{4} d^{5} g^{4}+3 B \,x^{4} b^{5} c \,d^{4} g^{4}+120 A \,x^{3} a^{2} b^{3} d^{5} g^{4}-16 B \,x^{3} a^{2} b^{3} d^{5} g^{4}+12 B \,b^{5} g^{4} c^{5}-36 B \,a^{4} b c \,d^{4} g^{4}-60 B \,a^{3} b^{2} c^{2} d^{3} g^{4}+90 B \,a^{2} b^{3} c^{3} d^{2} g^{4}-54 B a \,b^{4} c^{4} d \,g^{4}-4 B \,x^{3} b^{5} c^{2} d^{3} g^{4}+120 A \,x^{2} a^{3} b^{2} d^{5} g^{4}-36 B \,x^{2} a^{3} b^{2} d^{5} g^{4}+6 B \,x^{2} b^{5} c^{3} d^{2} g^{4}+60 A x \,a^{4} b \,d^{5} g^{4}-48 B x \,a^{4} b \,d^{5} g^{4}+48 B \,a^{5} d^{5} g^{4}+120 B \,x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{3} d^{5} g^{4}+20 B \,x^{3} a \,b^{4} c \,d^{4} g^{4}+120 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{3} b^{2} d^{5} g^{4}+60 B \,x^{2} a^{2} b^{3} c \,d^{4} g^{4}-30 B \,x^{2} a \,b^{4} c^{2} d^{3} g^{4}+60 B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{4} b \,d^{5} g^{4}-180 A \,a^{4} b c \,d^{4} g^{4}-60 A \,a^{5} d^{5} g^{4}+12 A \,x^{5} b^{5} d^{5} g^{4}+12 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{5} c^{5} g^{4}-12 B \ln \left (b x +a \right ) a^{5} d^{5} g^{4}+12 B \ln \left (b x +a \right ) b^{5} c^{5} g^{4}}{60 d^{5} b}\) | \(876\) |
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Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (168) = 336\).
Time = 0.35 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.41 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} g^{4} x^{5} - 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) + 3 \, {\left (B b^{5} c d^{4} + {\left (20 \, A - B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} - 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} - 2 \, {\left (15 \, A - 2 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} + 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} + 2 \, {\left (10 \, A - 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} - 12 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} - {\left (5 \, A - 4 \, B\right )} a^{4} b d^{5}\right )} g^{4} x + 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 12 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{60 \, b d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (155) = 310\).
Time = 3.78 (sec) , antiderivative size = 969, normalized size of antiderivative = 5.38 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {A b^{4} g^{4} x^{5}}{5} - \frac {B a^{5} g^{4} \log {\left (x + \frac {\frac {B a^{6} d^{5} g^{4}}{b} + 5 B a^{5} c d^{4} g^{4} - 10 B a^{4} b c^{2} d^{3} g^{4} + 10 B a^{3} b^{2} c^{3} d^{2} g^{4} - 5 B a^{2} b^{3} c^{4} d g^{4} + B a b^{4} c^{5} g^{4}}{B a^{5} d^{5} g^{4} + 5 B a^{4} b c d^{4} g^{4} - 10 B a^{3} b^{2} c^{2} d^{3} g^{4} + 10 B a^{2} b^{3} c^{3} d^{2} g^{4} - 5 B a b^{4} c^{4} d g^{4} + B b^{5} c^{5} g^{4}} \right )}}{5 b} + \frac {B c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) \log {\left (x + \frac {6 B a^{5} c d^{4} g^{4} - 10 B a^{4} b c^{2} d^{3} g^{4} + 10 B a^{3} b^{2} c^{3} d^{2} g^{4} - 5 B a^{2} b^{3} c^{4} d g^{4} + B a b^{4} c^{5} g^{4} - B a c g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right ) + \frac {B b c^{2} g^{4} \cdot \left (5 a^{4} d^{4} - 10 a^{3} b c d^{3} + 10 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d + b^{4} c^{4}\right )}{d}}{B a^{5} d^{5} g^{4} + 5 B a^{4} b c d^{4} g^{4} - 10 B a^{3} b^{2} c^{2} d^{3} g^{4} + 10 B a^{2} b^{3} c^{3} d^{2} g^{4} - 5 B a b^{4} c^{4} d g^{4} + B b^{5} c^{5} g^{4}} \right )}}{5 d^{5}} + x^{4} \left (A a b^{3} g^{4} - \frac {B a b^{3} g^{4}}{20} + \frac {B b^{4} c g^{4}}{20 d}\right ) + x^{3} \cdot \left (2 A a^{2} b^{2} g^{4} - \frac {4 B a^{2} b^{2} g^{4}}{15} + \frac {B a b^{3} c g^{4}}{3 d} - \frac {B b^{4} c^{2} g^{4}}{15 d^{2}}\right ) + x^{2} \cdot \left (2 A a^{3} b g^{4} - \frac {3 B a^{3} b g^{4}}{5} + \frac {B a^{2} b^{2} c g^{4}}{d} - \frac {B a b^{3} c^{2} g^{4}}{2 d^{2}} + \frac {B b^{4} c^{3} g^{4}}{10 d^{3}}\right ) + x \left (A a^{4} g^{4} - \frac {4 B a^{4} g^{4}}{5} + \frac {2 B a^{3} b c g^{4}}{d} - \frac {2 B a^{2} b^{2} c^{2} g^{4}}{d^{2}} + \frac {B a b^{3} c^{3} g^{4}}{d^{3}} - \frac {B b^{4} c^{4} g^{4}}{5 d^{4}}\right ) + \left (B a^{4} g^{4} x + 2 B a^{3} b g^{4} x^{2} + 2 B a^{2} b^{2} g^{4} x^{3} + B a b^{3} g^{4} x^{4} + \frac {B b^{4} g^{4} x^{5}}{5}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (168) = 336\).
Time = 0.23 (sec) , antiderivative size = 619, normalized size of antiderivative = 3.44 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {1}{5} \, A b^{4} g^{4} x^{5} + A a b^{3} g^{4} x^{4} + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, A a^{3} b g^{4} x^{2} + {\left (x \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {a \log \left (b x + a\right )}{b} + \frac {c \log \left (d x + c\right )}{d}\right )} B a^{4} g^{4} + 2 \, {\left (x^{2} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{3} b g^{4} + {\left (2 \, x^{3} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a^{2} b^{2} g^{4} + \frac {1}{6} \, {\left (6 \, x^{4} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a b^{3} g^{4} + \frac {1}{60} \, {\left (12 \, x^{5} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} + \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} + \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{4} g^{4} + A a^{4} g^{4} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 2030 vs. \(2 (168) = 336\).
Time = 0.47 (sec) , antiderivative size = 2030, normalized size of antiderivative = 11.28 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\text {Too large to display} \]
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Time = 1.62 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.60 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (B\,a^4\,g^4\,x+2\,B\,a^3\,b\,g^4\,x^2+2\,B\,a^2\,b^2\,g^4\,x^3+B\,a\,b^3\,g^4\,x^4+\frac {B\,b^4\,g^4\,x^5}{5}\right )-x^3\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,d}+\frac {A\,a\,b^3\,c\,g^4}{3\,d}\right )+x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{10\,b\,d}+\frac {a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}\right )+x\,\left (\frac {a^3\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{5\,b\,d}+\frac {2\,a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{b\,d}\right )+x^4\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{20\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (B\,a^4\,c\,d^4\,g^4-2\,B\,a^3\,b\,c^2\,d^3\,g^4+2\,B\,a^2\,b^2\,c^3\,d^2\,g^4-B\,a\,b^3\,c^4\,d\,g^4+\frac {B\,b^4\,c^5\,g^4}{5}\right )}{d^5}+\frac {A\,b^4\,g^4\,x^5}{5}-\frac {B\,a^5\,g^4\,\ln \left (a+b\,x\right )}{5\,b} \]
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