Integrand size = 38, antiderivative size = 85 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {B i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2} \]
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Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2562, 2341} \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]
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Rule 2341
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {A+B \log (e x)}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^3} \\ & = -\frac {B i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(85)=170\).
Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.45 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\frac {i \left (-\frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 (a+b x)^2}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)}-\frac {B d \left (\frac {1}{a+b x}+\frac {d \log (a+b x)}{b c-a d}-\frac {d \log (c+d x)}{b c-a d}\right )}{b^2}-\frac {B \left (\frac {b c-a d}{(a+b x)^2}-\frac {2 d}{a+b x}-\frac {2 d^2 \log (a+b x)}{b c-a d}+\frac {2 d^2 \log (c+d x)}{b c-a d}\right )}{4 b^2}\right )}{g^3} \]
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Time = 0.81 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.80
method | result | size |
parts | \(\frac {i A \left (-\frac {-a d +c b}{2 b^{2} \left (b x +a \right )^{2}}-\frac {d}{b^{2} \left (b x +a \right )}\right )}{g^{3}}-\frac {i B \,e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{g^{3} \left (a d -c b \right )}\) | \(153\) |
norman | \(\frac {\frac {B c d i x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a d -c b \right )}-\frac {2 A a d i +2 A b c i +B a d i +B b c i}{4 g \,b^{2}}-\frac {\left (2 A d i +B d i \right ) x}{2 g b}+\frac {B i \,c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}+\frac {B \,d^{2} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a d -c b \right ) g}}{\left (b x +a \right )^{2} g^{2}}\) | \(170\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e A}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\) | \(177\) |
default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e A}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\) | \(177\) |
parallelrisch | \(-\frac {-2 A \,c^{2} i \,b^{4} d +2 A \,a^{2} b^{2} d^{3} i +B \,a^{2} b^{2} d^{3} i -B \,b^{4} c^{2} d i -4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{2} i -2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{3} i +4 A x a \,b^{3} d^{3} i -4 A x \,b^{4} c \,d^{2} i +2 B x a \,b^{3} d^{3} i -2 B x \,b^{4} c \,d^{2} i -2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{2} d i}{4 g^{3} \left (b x +a \right )^{2} \left (a d -c b \right ) b^{4} d}\) | \(206\) |
risch | \(-\frac {B i \left (2 b d x +a d +c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g^{3} \left (b x +a \right )^{2} b^{2}}-\frac {i \left (2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+4 B \ln \left (d x +c \right ) a b \,d^{2} x -4 B \ln \left (-b x -a \right ) a b \,d^{2} x +4 A a b \,d^{2} x -4 A \,b^{2} c d x +2 B \ln \left (d x +c \right ) a^{2} d^{2}-2 B \,a^{2} \ln \left (-b x -a \right ) d^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-B \,b^{2} c^{2}\right )}{4 g^{3} \left (b x +a \right )^{2} b^{2} \left (a d -c b \right )}\) | \(249\) |
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Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (81) = 162\).
Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {2 \, {\left ({\left (2 \, A + B\right )} b^{2} c d - {\left (2 \, A + B\right )} a b d^{2}\right )} i x + {\left ({\left (2 \, A + B\right )} b^{2} c^{2} - {\left (2 \, A + B\right )} a^{2} d^{2}\right )} i + 2 \, {\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x + B b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (73) = 146\).
Time = 2.27 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.52 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=- \frac {B d^{2} i \log {\left (x + \frac {- \frac {B a^{2} d^{4} i}{a d - b c} + \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i - \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \log {\left (x + \frac {\frac {B a^{2} d^{4} i}{a d - b c} - \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i + \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {- 2 A a d i - 2 A b c i - B a d i - B b c i + x \left (- 4 A b d i - 2 B b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- B a d i - B b c i - 2 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (81) = 162\).
Time = 0.22 (sec) , antiderivative size = 570, normalized size of antiderivative = 6.71 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, B d i {\left (\frac {2 \, {\left (2 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{4} \, B c i {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {{\left (2 \, b x + a\right )} A d i}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {A c i}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B e^{3} i \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3}} + \frac {{\left (2 \, A e^{3} i + B e^{3} i\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
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Time = 2.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.32 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {x\,\left (2\,A\,b\,d\,i+B\,b\,d\,i\right )+A\,a\,d\,i+A\,b\,c\,i+\frac {B\,a\,d\,i}{2}+\frac {B\,b\,c\,i}{2}}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{2\,b^2\,g^3}+\frac {B\,a\,d\,i}{2\,b^3\,g^3}+\frac {B\,d\,i\,x}{b^2\,g^3}\right )}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]
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