Integrand size = 30, antiderivative size = 78 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {B (b c-a d) g x}{d}-\frac {B (b c-a d)^2 g \log (c+d x)}{b d^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 78, 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} \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g (a+b x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{2 b}-\frac {B g (b c-a d)^2 \log (c+d x)}{b d^2}+\frac {B g x (b c-a d)}{d} \]
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Rule 21
Rule 45
Rule 2548
Rubi steps \begin{align*} \text {integral}& = \frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}+\frac {(B (b c-a d)) \int \frac {(a g+b g x)^2}{(a+b x) (c+d x)} \, dx}{b g} \\ & = \frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}+\frac {(B (b c-a d) g) \int \frac {a+b x}{c+d x} \, dx}{b} \\ & = \frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b}+\frac {(B (b c-a d) g) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b} \\ & = \frac {B (b c-a d) g x}{d}-\frac {B (b c-a d)^2 g \log (c+d x)}{b d^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g \left (-\frac {2 B (-b c+a d) (b d x+(-b c+a d) \log (c+d x))}{d^2}+(a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right )}{2 b} \]
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Time = 0.65 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45
method | result | size |
risch | \(\frac {g B x \left (b x +2 a \right ) \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{2}+\frac {g b A \,x^{2}}{2}+g A a x -\frac {B \,a^{2} g \ln \left (b x +a \right )}{b}+\frac {2 g B \ln \left (-d x -c \right ) a c}{d}-\frac {g b B \ln \left (-d x -c \right ) c^{2}}{d^{2}}-g B a x +\frac {g b B c x}{d}\) | \(113\) |
derivativedivides | \(-\frac {-\frac {g A \left (b x +a \right )^{2}}{2}+g B \left (-\frac {\left (b x +a \right )^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{2}}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{b x +a}\right )}{d^{2}}+\frac {b x +a}{d}\right )\right )}{b}\) | \(142\) |
default | \(-\frac {-\frac {g A \left (b x +a \right )^{2}}{2}+g B \left (-\frac {\left (b x +a \right )^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{2}}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{b x +a}\right )}{d^{2}}+\frac {b x +a}{d}\right )\right )}{b}\) | \(142\) |
parts | \(A g \left (\frac {1}{2} b \,x^{2}+a x \right )-\frac {g B \left (-\frac {\left (b x +a \right )^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{2}}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{b x +a}\right )}{d^{2}}+\frac {b x +a}{d}\right )\right )}{b}\) | \(143\) |
parallelrisch | \(\frac {B \,x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{2} d^{2} g +A \,x^{2} b^{2} d^{2} g +2 B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a b \,d^{2} g +2 A x a b \,d^{2} g -2 B \ln \left (b x +a \right ) a^{2} d^{2} g +4 B \ln \left (b x +a \right ) a b c d g -2 B \ln \left (b x +a \right ) b^{2} c^{2} g -2 B x a b \,d^{2} g +2 B x \,b^{2} c d g +2 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a b c d g -B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{2} c^{2} g -2 A \,a^{2} d^{2} g -3 A a b c d g +2 B \,a^{2} d^{2} g -2 B \,b^{2} c^{2} g}{2 b \,d^{2}}\) | \(244\) |
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.91 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {A b^{2} d^{2} g x^{2} - 2 \, B a^{2} d^{2} g \log \left (b x + a\right ) + 2 \, {\left (B b^{2} c d + {\left (A - B\right )} a b d^{2}\right )} g x - 2 \, {\left (B b^{2} c^{2} - 2 \, B a b c d\right )} g \log \left (d x + c\right ) + {\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{2 \, b d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (68) = 136\).
Time = 0.92 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.21 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {A b g x^{2}}{2} - \frac {B a^{2} g \log {\left (x + \frac {\frac {B a^{3} d^{2} g}{b} + 2 B a^{2} c d g - B a b c^{2} g}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{b} + \frac {B c g \left (2 a d - b c\right ) \log {\left (x + \frac {3 B a^{2} c d g - B a b c^{2} g - B a c g \left (2 a d - b c\right ) + \frac {B b c^{2} g \left (2 a d - b c\right )}{d}}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{d^{2}} + x \left (A a g - B a g + \frac {B b c g}{d}\right ) + \left (B a g x + \frac {B b g x^{2}}{2}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (76) = 152\).
Time = 0.21 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.21 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{2} \, A b g x^{2} + {\left (x \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a g + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B b g + A a g x \]
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Time = 0.70 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.69 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{2} \, A b g x^{2} - \frac {B a^{2} g \log \left (b x + a\right )}{b} + \frac {1}{2} \, {\left (B b g x^{2} + 2 \, B a g x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {{\left (B b c g + A a d g - B a d g\right )} x}{d} - \frac {{\left (B b c^{2} g - 2 \, B a c d g\right )} \log \left (-d x - c\right )}{d^{2}} \]
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Time = 1.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.54 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=x\,\left (\frac {g\,\left (2\,A\,a\,d+A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}-\frac {A\,g\,\left (a\,d+b\,c\right )}{d}\right )+\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (\frac {B\,b\,g\,x^2}{2}+B\,a\,g\,x\right )+\frac {A\,b\,g\,x^2}{2}-\frac {B\,a^2\,g\,\ln \left (a+b\,x\right )}{b}+\frac {B\,c\,g\,\ln \left (c+d\,x\right )\,\left (2\,a\,d-b\,c\right )}{d^2} \]
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