Integrand size = 32, antiderivative size = 138 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {B}{2 b g^3 (a+b x)^2}+\frac {B d}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}-\frac {B d^2 \log (c+d x)}{b (b c-a d)^2 g^3} \]
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Time = 0.06 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2548, 21, 46} \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{2 b g^3 (a+b x)^2}+\frac {B d^2 \log (a+b x)}{b g^3 (b c-a d)^2}-\frac {B d^2 \log (c+d x)}{b g^3 (b c-a d)^2}+\frac {B d}{b g^3 (a+b x) (b c-a d)}-\frac {B}{2 b g^3 (a+b x)^2} \]
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Rule 21
Rule 46
Rule 2548
Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (a g+b g x)^2} \, dx}{b g} \\ & = -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b g^3} \\ & = -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3} \\ & = -\frac {B}{2 b g^3 (a+b x)^2}+\frac {B d}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}-\frac {B d^2 \log (c+d x)}{b (b c-a d)^2 g^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+\frac {B \left ((b c-a d) (-3 a d+b (c-2 d x))-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )}{(b c-a d)^2}}{2 b g^3 (a+b x)^2} \]
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Time = 1.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.73
method | result | size |
parallelrisch | \(-\frac {2 B x a \,b^{4} d^{3}-2 B x \,b^{5} c \,d^{2}-B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} d^{3}+A \,a^{2} b^{3} d^{3}+A \,b^{5} c^{2} d +3 B \,a^{2} b^{3} d^{3}+B \,b^{5} c^{2} d -2 A a \,b^{4} c \,d^{2}-4 B a \,b^{4} c \,d^{2}+B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} c^{2} d -2 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} d^{3}-2 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} c \,d^{2}}{2 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} d}\) | \(239\) |
risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 b \,g^{3} \left (b x +a \right )^{2}}-\frac {2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+4 B \ln \left (d x +c \right ) a b \,d^{2} x -4 B \ln \left (-b x -a \right ) a b \,d^{2} x +2 B \ln \left (d x +c \right ) a^{2} d^{2}-2 B \,a^{2} \ln \left (-b x -a \right ) d^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +A \,a^{2} d^{2}-2 A a b c d +A \,b^{2} c^{2}+3 B \,a^{2} d^{2}-4 B a b c d +B \,b^{2} c^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g^{3} \left (b x +a \right )^{2} b}\) | \(244\) |
parts | \(-\frac {A}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {\left (2 B a d -B b c \right ) x}{a g \left (a d -c b \right )}+\frac {B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {B c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (3 B a d -B b c \right ) b \,x^{2}}{2 g \,a^{2} \left (a d -c b \right )}+\frac {B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}}{g^{2} \left (b x +a \right )^{2}}\) | \(251\) |
norman | \(\frac {\frac {\left (A a d -A b c +2 B a d -B b c \right ) x}{a g \left (a d -c b \right )}+\frac {B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {B c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (A a d -A b c +3 B a d -B b c \right ) b \,x^{2}}{2 a^{2} g \left (a d -c b \right )}+\frac {B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}}{g^{2} \left (b x +a \right )^{2}}\) | \(252\) |
derivativedivides | \(-\frac {\frac {d^{3} A \left (\frac {b}{2 \left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}-\frac {1}{\left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}\right )}{g^{3}}+\frac {\frac {B \,d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}+\frac {3 B \,d^{3}}{2 b g \left (d x +c \right )^{2}}-\frac {b B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\) | \(286\) |
default | \(-\frac {\frac {d^{3} A \left (\frac {b}{2 \left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}-\frac {1}{\left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}\right )}{g^{3}}+\frac {\frac {B \,d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}+\frac {3 B \,d^{3}}{2 b g \left (d x +c \right )^{2}}-\frac {b B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\) | \(286\) |
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Time = 0.27 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {{\left (A + B\right )} b^{2} c^{2} - 2 \, {\left (A + 2 \, B\right )} a b c d + {\left (A + 3 \, B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x - B b^{2} c^{2} + 2 \, B a b c d\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (122) = 244\).
Time = 1.03 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.03 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{2 a^{2} b g^{3} + 4 a b^{2} g^{3} x + 2 b^{3} g^{3} x^{2}} - \frac {B d^{2} \log {\left (x + \frac {- \frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} + \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {B d^{2} \log {\left (x + \frac {\frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} - \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {- A a d + A b c - 3 B a d + B b c - 2 B b d x}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (134) = 268\).
Time = 0.20 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.22 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=\frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {A}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=\frac {B d^{2} \log \left (b x + a\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac {B d^{2} \log \left (d x + c\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac {B \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} + \frac {2 \, B b d x - A b c - B b c + A a d + 3 \, B a d}{2 \, {\left (b^{4} c g^{3} x^{2} - a b^{3} d g^{3} x^{2} + 2 \, a b^{3} c g^{3} x - 2 \, a^{2} b^{2} d g^{3} x + a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )}} \]
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Time = 1.95 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {\frac {A\,a\,d-A\,b\,c+3\,B\,a\,d-B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b\,d\,x}{a\,d-b\,c}}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {2\,B\,d^2\,\mathrm {atanh}\left (\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]
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