Integrand size = 38, antiderivative size = 85 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=-\frac {B g (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) i^3 (c+d x)^2} \]
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Time = 0.04 (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} \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=\frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
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Rule 2341
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {g \text {Subst}\left (\int x (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) i^3} \\ & = -\frac {B g (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) i^3 (c+d x)^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(85)=170\).
Time = 0.10 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.44 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=\frac {g \left (\frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^2 (c+d x)^2}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 (c+d x)}+\frac {b B \left (\frac {1}{c+d x}+\frac {b \log (a+b x)}{b c-a d}-\frac {b \log (c+d x)}{b c-a d}\right )}{d^2}-\frac {B \left (\frac {b c-a d}{(c+d x)^2}+\frac {2 b}{c+d x}+\frac {2 b^2 \log (a+b x)}{b c-a d}-\frac {2 b^2 \log (c+d x)}{b c-a d}\right )}{4 d^2}\right )}{i^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(81)=162\).
Time = 0.81 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05
method | result | size |
norman | \(\frac {-\frac {2 A a d g +2 A b c g -B a d g -B b c g}{4 i \,d^{2}}-\frac {\left (2 A b g -B b g \right ) x}{2 i d}-\frac {B \,a^{2} g \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a d -c b \right )}-\frac {B \,b^{2} g \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a d -c b \right ) i}-\frac {B a b g x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a d -c b \right )}}{i^{2} \left (d x +c \right )^{2}}\) | \(174\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\) | \(181\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\) | \(181\) |
parallelrisch | \(-\frac {-B \,a^{2} b \,d^{4} g +B \,b^{3} c^{2} d^{2} g +4 A x a \,b^{2} d^{4} g -4 A x \,b^{3} c \,d^{3} g -2 B x a \,b^{2} d^{4} g +2 B x \,b^{3} c \,d^{3} g +2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{4} g +2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4} g +2 A \,a^{2} b \,d^{4} g -2 A \,b^{3} c^{2} d^{2} g +4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4} g}{4 i^{3} \left (d x +c \right )^{2} \left (a d -c b \right ) b \,d^{4}}\) | \(206\) |
risch | \(-\frac {B g \left (2 b d x +a d +c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 d^{2} i^{3} \left (d x +c \right )^{2}}-\frac {g \left (-2 B \ln \left (-d x -c \right ) b^{2} d^{2} x^{2}+2 B \ln \left (b x +a \right ) b^{2} d^{2} x^{2}-4 B \ln \left (-d x -c \right ) b^{2} c d x +4 B \ln \left (b x +a \right ) b^{2} c d x +4 A a b \,d^{2} x -4 A \,b^{2} c d x -2 B \ln \left (-d x -c \right ) b^{2} c^{2}+2 B \ln \left (b x +a \right ) b^{2} c^{2}-2 B a b \,d^{2} x +2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-2 A \,b^{2} c^{2}-B \,a^{2} d^{2}+B \,b^{2} c^{2}\right )}{4 d^{2} i^{3} \left (d x +c \right )^{2} \left (a d -c b \right )}\) | \(249\) |
parts | \(\frac {g A \left (-\frac {b}{d^{2} \left (d x +c \right )}-\frac {a d -c b}{2 d^{2} \left (d x +c \right )^{2}}\right )}{i^{3}}-\frac {g B d \left (a \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )-\frac {c b \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{d}\right )}{i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(255\) |
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.18 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=-\frac {2 \, {\left ({\left (2 \, A - B\right )} b^{2} c d - {\left (2 \, A - B\right )} a b d^{2}\right )} g x + {\left ({\left (2 \, A - B\right )} b^{2} c^{2} - {\left (2 \, A - B\right )} a^{2} d^{2}\right )} g - 2 \, {\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x + B a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (71) = 142\).
Time = 2.32 (sec) , antiderivative size = 382, normalized size of antiderivative = 4.49 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=\frac {B b^{2} g \log {\left (x + \frac {- \frac {B a^{2} b^{2} d^{2} g}{a d - b c} + \frac {2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g - \frac {B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac {B b^{2} g \log {\left (x + \frac {\frac {B a^{2} b^{2} d^{2} g}{a d - b c} - \frac {2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g + \frac {B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac {- 2 A a d g - 2 A b c g + B a d g + B b c g + x \left (- 4 A b d g + 2 B b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac {\left (- B a d g - B b c g - 2 B b d g x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (81) = 162\).
Time = 0.22 (sec) , antiderivative size = 567, normalized size of antiderivative = 6.67 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=-\frac {1}{4} \, B b g {\left (\frac {2 \, {\left (2 \, d x + c\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}} - \frac {b c^{2} - 3 \, a c d + 2 \, {\left (b c d - 2 \, a d^{2}\right )} x}{{\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}} - \frac {2 \, {\left (b^{2} c - 2 \, a b d\right )} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}} + \frac {2 \, {\left (b^{2} c - 2 \, a b d\right )} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}}\right )} + \frac {1}{4} \, B a g {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac {{\left (2 \, d x + c\right )} A b g}{2 \, {\left (d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}\right )}} - \frac {A a g}{2 \, {\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \]
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Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} {\left (\frac {2 \, {\left (b e x + a e\right )}^{2} B g \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2} e i^{3}} + \frac {{\left (b e x + a e\right )}^{2} {\left (2 \, A g - B g\right )}}{{\left (d x + c\right )}^{2} e i^{3}}\right )} \]
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Time = 2.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.33 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^3} \, dx=-\frac {x\,\left (2\,A\,b\,d\,g-B\,b\,d\,g\right )+A\,a\,d\,g+A\,b\,c\,g-\frac {B\,a\,d\,g}{2}-\frac {B\,b\,c\,g}{2}}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,a\,g}{2\,d^2\,i^3}+\frac {B\,b\,c\,g}{2\,d^3\,i^3}+\frac {B\,b\,g\,x}{d^2\,i^3}\right )}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {B\,b^2\,g\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \]
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