Integrand size = 40, antiderivative size = 173 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {b B (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}+\frac {B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^2 g^2 i}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i} \]
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Time = 0.11 (sec) , antiderivative size = 173, 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, 14, 2338} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {b B (c+d x)}{g^2 i (a+b x) (b c-a d)^2}+\frac {B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^2 i (b c-a d)^2} \]
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Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x) (A+B \log (e x))}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = -\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {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)^2 g^2 i}-\frac {B \text {Subst}\left (\int \frac {-b-d x \log (x)}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = -\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {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)^2 g^2 i}-\frac {B \text {Subst}\left (\int \left (-\frac {b}{x^2}-\frac {d \log (x)}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = -\frac {b B (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {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)^2 g^2 i}+\frac {(B d) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = -\frac {b B (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}+\frac {B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^2 g^2 i}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {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)^2 g^2 i} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.69 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{2 (b c-a d)^2 g^2 i (a+b x)} \]
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Time = 0.98 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.33
method | result | size |
parts | \(\frac {A \left (\frac {d \ln \left (d x +c \right )}{\left (a d -c b \right )^{2}}+\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}-\frac {d \ln \left (b x +a \right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i}-\frac {B \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i d}\) | \(230\) |
norman | \(\frac {-\frac {\left (A a d +B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B a d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b \left (A d +B d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (A +B \right ) b x}{g i a \left (a d -c b \right )}-\frac {b B d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g \left (b x +a \right )}\) | \(252\) |
parallelrisch | \(-\frac {2 A x \,a^{3} b \,c^{2} d +2 B x \,a^{3} b \,c^{2} d +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} c^{2} d -2 A x \,a^{2} b^{2} c^{3}+2 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} c^{2} d -2 B x \,a^{2} b^{2} c^{3}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +2 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d}{2 i \,g^{2} \left (b x +a \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) c^{2} a^{3}}\) | \(254\) |
risch | \(\frac {A d \ln \left (d x +c \right )}{g^{2} i \left (a d -c b \right )^{2}}+\frac {A}{g^{2} i \left (b x +a \right ) \left (a d -c b \right )}-\frac {A d \ln \left (b x +a \right )}{g^{2} i \left (a d -c b \right )^{2}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{2} i \left (a d -c b \right )^{2}}-\frac {B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\) | \(266\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\) | \(288\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\) | \(288\) |
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Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {2 \, {\left (A + B\right )} b c - 2 \, {\left (A + B\right )} a d + {\left (B b d x + B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A + B\right )} b d x + B b c + A a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (144) = 288\).
Time = 0.62 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.23 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=- \frac {B d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g^{2} i - 4 a b c d g^{2} i + 2 b^{2} c^{2} g^{2} i} + \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A + B\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (171) = 342\).
Time = 0.22 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.45 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=-B {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - A {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} + \frac {{\left ({\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \, {\left (a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x\right )}} \]
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\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}} \,d x } \]
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Time = 2.45 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.39 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)} \, dx=\frac {A+B}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-\frac {B\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+B\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \]
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